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REGIONAL MATHEMATICS CURRICULUM GUIDE DEVELOPED BY AND FOR: THE REGIONAL DISTRICTS OF FRANKFORD, LAFAYETTE, AND SUSSEX-WANTAGE SCHOOLS 2008 TABLE OF CONTENTS Page Credits………………………………………………………………3 Philosophy…………………………………………………………..4 Implementation……………………………………………………...5 Assessment……………………………………………….…………6 Standards……………………………………………………………7 Instructional Units (Pre-School – Grade 8)………………………..12 Appendix NJ CCCS Mathematics Revisions…………………………..187 Standards 4.1 – 4.5: Vertical Alignment…………………...190 Internet Websites……………………………………………221 2 CREDITS Grateful recognition is made to the following individuals for their level of expertise and dedicated work. MEMBERS: Kim Branham – Lafayette June Yucius – Frankford Tara MacGlashan – Lafayette Carole Flynn – Frankford Allyn Perry – Lafayette Cathy Gardner – Frankford Linda Jensen – Lafayette Jane Fialcowitz – Frankford Nicole Worthington – Lafayette Judy Gray – Frankford Anne Lajeunesse – Lafayette Linda Suchana – Frankford Carol Wilson – Sussex-Wantage Jodi Kuzmiak – Frankford Joan Casserly – Sussex-Wantage Darcy Harris – Frankford Cathryn Weiss – Sussex-Wantage Sara Beattie – Frankford Maureen Vatalaro – Sussex-Wantage Appreciation to the following members for their organization and guidance: Genene Pagliaro – Frankford / Lafayette Susan Petrick – Sussex-Wantage 3 PHILOSOPHY All students should be afforded the opportunity to achieve mathematics proficiency through an integration of understanding, comprehending, applying, reasoning, and analyzing. The math opportunity for students must be rich and complex and connected to real life experiences. Through a specific math language, students will be able to communicate comprehension in both oral and written form. We must enable all of our children to acquire math skills, understanding, and attitudes that they will need to be successful in their careers and daily lives. We believe all students can learn math and all students need to learn math. The local community, educators, parents, and students shall work together to make the revision of the NJ Mathematics Standards a reality. 4 IMPLEMENTATION The New Jersey Core Curriculum Content Standards provided the basis for the development of this curriculum guide. In order for implementation of this curriculum to occur, the following need to be in place: 1. Adequate instructional time; 2. Infusion of mathematics concepts in all content areas; 3. Effective instructional materials, including all necessary supplemental and technological support and resources; 4. Consistent and sustained Professional Development for all teachers of mathematics. 5 ASSESSMENT THE ASSESSMENT PROCESS: Assessment is a way of providing feedback to the various stakeholders in the educational system, and of communicating the outcomes to all concerned. The data provide feedback to: students on how well they are meeting expectations teachers with how well students are learning districts on the effectiveness of their programs policy makers on how well policies are working. CLASSROOM ASSESSMENT: Classroom teachers should utilize a variety of assessment tools designed to provide information on student comprehension and progress toward learning objectives. Assessment should be based upon, but not limited to, the following: 1. Open-ended problems 2. Teacher interviews 3. Portfolios 4. Mathematical journals 5. Formative and summative assessments 6. Completion of assignments, both in and out of the classroom 7. Oral contribution in class 8. Rubrics 6 Mathematics Standards 2002 The revised standards adopted in 2002 are more specific and clearer than the 1996 standards. The 16 standards of 1996 have been organized into 5 standards that correspond to the content clusters of the statewide assessments. There are student expectations at each grade level beginning with kindergarten. Standards 1 – 4 define the content that students should know and be able to do. 4.1. Number and Numerical Operations A. Number Sense B. Numerical Operations C. Estimation 4.2. Geometry and Measurement A. Geometric Properties B. Transforming Shapes C. Coordinate Geometry D. Units of Measurement E. Measuring Geometric Objects 4.3. Patterns and Algebra A. Patterns and Relationships B. Functions C. Modeling D. Procedures 4.4. Data Analysis, Probability, and Discrete Mathematics A. Data Analysis (Statistics) B. Probability C. Discrete Mathematics—Systematic Listing and Counting D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms The fifth standard defines the mathematical processes that all students at each grade level should use when acquiring and applying content knowledge and skills of the first four standards. 7 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 8 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 9 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 10 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 11 MATHEMATICS PRE-SCHOOL EXPECTATION STUDENT OUTCOME TEACHER’S NOTES AND SUPPLEMENTARY RESOURCES EXPECTATION 1: 1.1 Demonstrates understanding of one-to-one Children demonstrate an correspondence (e.g., places one placemat at each place, understanding of number and gives each child one cookie, places one animal in each numerical operations. truck, hands out manipulatives to be shared with a friend saying "One for you, one for me."). 1.2 Spontaneously counts for own purposes (e.g., counting blocks or cars, counting beads while stringing them, handing out napkins). 1.3 Learns to say the counting numbers. 1.4 Discriminates numbers from other symbols in the environment (e.g., street signs, license plates, room number, clock, etc.). 1.5 Recognizes and names some written numerals. 1.6 Compares numbers in different contexts (e.g., using words such as more and less). 1.7 Uses estimation as a method for approximating an appropriate amount (e.g., at snack time, deciding how many napkins to take from a large pile for the group, determining number of blocks to use when building structures). 12 1.8 Adds two groups of concrete objects by counting the total (e.g., three blue pegs, three yellow pegs, six pegs altogether). 1.9 Subtracts one group of concrete objects from another by taking some away and then counting the remainder (e.g., "I have four carrot sticks. I'm eating one! Now I have 3!"). EXPECTATION 2: 2.1 Identifies basic shapes in the environment (e.g., circle, Children develop knowledge of square, triangle, cube, sphere). spatial concepts, e.g., shapes and measurement. 2.2 Uses standard and nonstandard measurement units (e.g., measuring body length with unifix cubes, using a tape measure to gauge height of block construction, counting the number of cups it takes to fill a bucket with water). 2.3 Uses vocabulary to describe distances (e.g., "It was a really long walk to the playground."). 2.4 Uses vocabulary to describe directional concept (e.g., "Watch me climb up the ladder and slide down."). 2.5 Uses positional words in a functional way (e.g., "I put the red block on top of the cabinet."). 2.6 Makes three-dimensional constructions and models (e.g., sculptures that have height, depth and width). 2.7 Makes connections between two dimensional and three dimensional forms (e.g., circle-sphere, square-cube, triangle-pyramid). EXPECTATION 3: 3.1 Sorts objects into groups (e.g., separate basket of Children understand patterns, collected items into piles of pinecones, acorns and twigs). relationships and classification. 13 3.2 Classifies objects by sorting them into subgroups by one or more attributes (e.g., sorting counting bears by color into trays, separating a mixture of beans by individual size and shape). 3.3 Describes an object by characteristics it does or does not possess (e.g., "This button doesn't have holes."). 3.4 Seriates objects according to various properties including size, number, length, heaviness, texture (rough to smooth) or loudness. 3.5 Identifies patterns in the environment (e.g., "Look at the rug. It has a circle, then a number, then a letter..."). 3.6 Represents patterns in a variety of ways (e.g., stringing beads red/green/red/green/red/green, arranging buttons big/bigger/biggest, or singing songs that follow a simple pattern). EXPECTATION 4: 4.1 Starts and stops on a signal (e.g., freezing in position Children develop knowledge of when the music stops). sequence and temporal awareness. 4.2 Describes the sequence of the daily routine and demonstrates understanding of basic temporal relations (e.g., "We will go outside after snack time."). 4.3 Arranges pictures of events in temporal order (e.g., first, a photo of the child eating breakfast; second, a photo of the child getting on the bus; third, a photo of the child in the classroom). 14 EXPECTATION 5: 5.1 Uses mathematical terms when conversing with Children will use mathematical others (e.g., "Which car is faster?" "My building is knowledge to represent, communicate taller than yours." "I have more sand in my and solve problems in their environment. bucket."). 5.2 Uses emergent mathematical knowledge as a problem-solving tool (e.g., Maritza notices that Juan has more carrot sticks than she does. She says, "May I have some of yours? Then we will have the same amount." Jorge decides to fill his bucket by using small cups of water when he realizes that he cannot fit the bucket under the faucet). 5.3 Describes how he/she solved mathematical problems in his/her own way. 15 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 16 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 17 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 18 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 19 MATHEMATICS KINDERGARTEN TEACHER’S NOTES & STANDARD STUDENT OUTCOME SUGGESTED ACTIVITIES SUPPLEMENTARY NJ FRAMEWORKS 1996 RESOURCES 4.1 All students will develop 4.1.A. Number Sense Pgs. 176-177, Overview number sense and will perform standard numerical 4.1.A.1 – Use real-life Pg. 178, No. 1: Use real-life operations and estimations experiences, physical experiences, physical materials, and on all types of numbers in a materials, and technology to technology to construct meanings for variety of ways. construct meanings for whole numbers, commonly used 4.1.A. Number Sense numbers. fractions, and decimals. 4.1.B. Numerical Operations • Whole numbers to 20 4.1.C. Estimation • Ordinals to 10 • Proper fractions (denominators of 2; concept of 1/2) 4.1.A.2 – Demonstrate an Pg. 179, No. 4: Develop a sense of the understanding of whole magnitudes of whole numbers and number place value concepts commonly used fractions. (tens and ones—units). 4.1.A.3 – Understand that Pg. 180, No. 5: Understand the numbers have a variety of uses. various uses of numbers including counting, measuring, labeling, and indicating location. 4.1.A.4 – Count and perform simple computations with coins. • Amounts up to $1.00 (using cents notation) (Recognition of four basic 20 coins) 4.1.A.5 – Compare and order Pg. 181, No. 8: Compare and order whole numbers. whole numbers, commonly used fractions, and decimals. 4.1.B. Numerical Operations Pgs. 253-255, Overview 4.1.B.1 – Develop the meanings Pg. 256, No. 1: Develop meaning for of addition and subtraction by the four basic arithmetic operations by concretely modeling and modeling and discussing a variety of discussing a large variety of problems. problems. Pg. 257, No. 2: Develop proficiency • Joining, separating, and with and memorize basic number facts comparing using a variety of fact strategies (such as “counting on” and “doubles”). Pg. 257, No. 3: Construct, use and explain procedures for performing whole number calculations in the various methods of computation. Pg. 258, No. 5: Use a variety of mental computation and estimation techniques. Pg. 259, No. 6: Select and use appropriate computational methods from mental math, estimation, paper- and-pencil, and calculator methods, and check, the reasonableness of results. Pg. 259, No. 7: Understand and use relationships among operations and properties of operations. 21 4.1.C. Estimation Pgs. 300-310, Overview 4.1.C.1 – Judge without Pg. 312, No. 1: Judge without counting whether a set of counting whether a set of objects has objects has less than, more less than, more than, or the same than, or the same number of number of objects as a reference set. objects as a reference set. 4.1.C.3 – Explore a variety of Pg. 313, No. 4: Explore, construct, strategies for estimating both and use a variety of estimation quantities (e.g., the number of strategies. marbles in a jar) and results of Pg. 258, No. 5: Use a variety of computation. mental computation and estimation techniques. Pg. 259, No. 6: Select and use appropriate computational methods from mental math, estimation, paper- and-pencil, and calculator methods, and check the reasonableness of results. Pg. 288, No. 6: Understand and incorporate estimation and repeated measures in measurement activities. 4.2 All students will develop 4.2.A. Geometric Properties Pgs. 213-214, Overview spatial sense and the ability to use geometric properties, 4.2.A.1 – Identify and describe Pg. 215, No. 1: Explore spatial relationships, and spatial relationships among relationships such as the direction, measurement to model, objects in space and their orientation, and perspectives of objects describe and analyze relative shapes and sizes. in space, their relative shapes and phenomena. • Inside/outside, sizes, and the relations between objects 4.2.A. Geometric Properties left/right, above/below, and their shadows or projections. 4.2.B. Transforming Shapes between 4.2.D. Units of Measurement • Smaller/larger/same 22 size, wider/narrower, longer/shorter • Congruence (i.e. same size and shape) 4.2.A.2 – Use concrete objects, Pg. 215, No. 3: Explore properties of drawings, and computer three- and two-dimensional shapes graphics to identify, classify, using concrete objects, drawings, and and describe standard three- computer graphics. dimensional and two- Pg. 216, No. 4: Use properties of dimensional shapes. three- and two-dimensional shapes to • 2D figures – square, identify, classify, and describe shapes. rectangle, circle, triangle 4.2.A.4 – Recognize, describe, Pg. 217, No. 7: Explore geometric extend and create designs and transformations such as rotations patterns with geometric (turns), reflections (flips), and objects of different shapes and translations (slides). colors. Pg. 217, No. 9: Understand the variety of ways in which geometric shapes and objects can be measured. Pg. 217, No. 10: Investigate the occurrence of geometry in nature, art, and other areas. Pg. 216, No. 6: Use tessellations to explore properties of geometric shapes and their relationships to the concepts of area and perimeter. 4.2.B. Transforming Shapes Pgs. 338-339, Overview 4.2.B.1 – Use simple shapes to Pg. 340, No. 1: Reproduce, extend, make designs, patterns and create, and describe patterns and 23 pictures. sequences using a variety of materials. Pg. 342, No. 5: Observe and recognize examples of patterns, relationships, and functions in other disciplines and contexts. Pg. 343, No. 6: Form and verify generalizations based on observations of patterns and relationships. Pg. 216, No. 6: Use tessellations to explore properties of geometric shapes and their relationships to the concepts of area and perimeter. Pg. 217, No. 9: Understand the variety of ways in which geometric shapes and objects can be measured. Pg. 217, No. 10: Investigate the occurrence of geometry in nature, art, and other areas. 4.2.D. Units of Measurement Pgs. 284-285, Overview 4.2.D.1 – Directly compare and Pg. 287, No. 2: Compare and order order objects according to objects according to some measurable measurable attributes. attribute. • Attributes – length, Pg. 288, No. 4: Develop and use weight, capacity, time, personal referents for standard units of temperature measure (such as the width of a finger to approximate a centimeter). 4.2.D.2 – Recognize the need Pg. 288, No. 3: Recognize the need for a uniform unit of for a uniform unit of measure. measurement. 24 4.2.D.4 – Estimate measures. Pg. 288, No. 6: Understand and incorporate estimation and repeated measures in measurement activities. 4.3 All students will represent 4.3.A. Patterns Pgs. 338-339, Overview and analyze relationships among variable quantities 4.3.A.1 – Recognize, describe, Pg. 340, No. 1: Reproduce, extend, and solve problems extend, and create patterns. create, and describe patterns and involving patterns, • Using concrete sequences using a variety of materials. functions, and algebraic materials Pg. 341, No. 2: Use tables, rules, concepts and processes. (manipulatives), variables, open sentences, and graphs 4.3.A. Patterns pictures, rhythms, and to describe patterns and other 4.3.C. Modeling whole numbers. relationships. • Descriptions using words and symbols (e.g., “add two” or “+2”) 4.3.C. Modeling Pg. 490, Overview 4.3.C.1 – Recognize and Pg. 491, No. 2: Investigate and describe changes over time describe how certain quantities change (e.g., temperature, height). over time. 4.4 All students will develop an 4.4.C. Discrete Mathematics – Pgs. 445-447, Overview understanding of the Systematic Listing and concepts and techniques of Counting data analysis, probability, and discrete mathematics, 4.4.C.1 – Sort and classify Pg. 449, No. 4: Investigate ways to and will use them to model objects according to attributes. represent and classify data according situations, solve problems, • Venn diagrams to attributes, such as shape or color, and analyze and draw and relationships, and discuss the appropriate inferences from purpose and usefulness of such data. classification. 4.4.C. Discrete Mathematics – Systematic Listing and 25 Counting 4.4.D. Discrete Mathematics – Pgs. 445-447, Overview 4.4.D. Discrete Mathematics – Vertex-Edge Graphs and Vertex-Edge Graphs and Algorithms Algorithms 4.4.D.1 – Follow simple sets of Pg. 450, No. 5: Follow, devise, and directions (e.g., from one describe practical lists of instructions. location to another, or from a Pg. 448, No. 2: Use networks and tree recipe). diagrams to represent everyday situations. 4.4.D.2 – Color simple maps Pg. 448, No. 2: Use networks and tree with a small number of colors. diagrams to represent everyday situations. 26 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 27 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 28 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 29 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 30 MATHEMATICS FIRST GRADE STANDARD STUDENT OUTCOME SUGGESTED ACTIVITIES TEACHER’S NOTES AND NJ FRAMEWORKS 1996 SUPPLEMENTARY RESOURCES 4.1 All students will develop 4.1.A. Number Sense Pgs. 176-177, Overview number sense and will perform standard numerical 4.1.A.1 – Use real-life Pg. 178, No. 1: Use real-life operations and estimations experiences, physical materials, experiences, physical materials, on all types of numbers in a and technology to construct and technology to construct variety of ways. meanings for numbers. meanings for whole numbers, 4.1.A. Number Sense • Whole numbers through commonly used fractions, and 4.1.B. Numerical Operations hundreds decimals. 4.1.C. Estimation • Ordinals • Proper fractions (denominators of 2, 3, 4) 4.1.A.2 – Demonstrate an Pg. 179, No. 4: Develop a sense understanding of whole number of the magnitudes of whole place value concepts. numbers, commonly used fractions, and decimals. 4.1.A.3 – Understand that Pg. 180, No. 5: Understand the numbers have a variety of uses. various uses of numbers including counting, measuring, labeling, and indicating location. 4.1.A.4 – Count and perform Pg. 180, No. 6: Count and simple computations with perform simple computations coins. with money. • Amounts up to $1.00 (using cents notation) 31 4.1.A.5 – Compare and order Pg. 181, No. 8: Compare and whole numbers. order whole numbers, commonly used fractions, and decimals. 4.1.B. Numerical Operations Pgs. 253-255, Overview 4.1.B.1 – Develop the meanings Pg. 256, No. 1: Develop of addition and subtraction by meaning for the four basic concretely modeling and arithmetic operations by discussing a large variety of modeling and discussing a problems. variety of problems. • Joining, separating, and Pg. 257, No. 2: Develop comparing proficiency with and memorize basic number facts using a variety of fact strategies (such as “counting on” and “doubles”). Pg. 257, No. 3: Construct, use and explain procedures for performing whole number calculations in the various methods of computation. Pg. 258, No. 5: Use a variety of mental computation and estimation techniques. Pg. 259, No. 6: Select and use appropriate computational methods from mental math, estimation, paper-and-pencil, and calculator methods, and check, the reasonableness of results. Pg. 259, No. 7: Understand and use relationships among 32 operations and properties of operations. 4.1.B.3 – Develop proficiency Pg. 257, No. 2: Develop with basic addition and proficiency with and memorize subtraction number facts using basic number facts using a a variety of fact strategies variety of fact strategies (such (such as “counting on” and as “counting on” and “near doubles”) and then “doubles”). commit them to memory. 4.1.B.4 – Construct, use, and Pg. 257, No. 3: Construct, use, explain procedures for and explain procedures for performing addition and performing whole number subtraction calculations with: calculations in the various • Pencil-and-paper methods of computation. • Mental math • Calculator 4.1.B.5 – Use efficient and Pg. 259, No. 6: Select and use accurate pencil-and-paper appropriate computational procedures for computation methods from mental math, with whole numbers. estimation, paper-and-pencil, • Addition of 2-digit and calculator methods, and numbers check the reasonableness of • Subtraction of 2-digit results. numbers 4.1.B.6 – Select pencil-and- Pg. 259, No. 6: Select and use paper, mental math, or a appropriate computational calculator as the appropriate methods from mental math, computational method in a estimation, paper-and-pencil, given situation depending on and calculator methods, and the context and numbers. check the reasonableness of results. 33 4.1.B.7 – Check the Pg. 259, No. 6: Select and use reasonableness of results of appropriate computational computations. methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results. Pg. 258, No. 5: Use a variety of mental computation and estimation techniques. 4.1.B.8 – Understand and use Pg. 259, No. 7: Understand and the inverse relationship use relationships among between addition and operations and properties of subtraction. operations. 4.1.C. Estimation Pg. 311, Overview 4.1.C.1 – Judge without counting Pg. 312, No. 1: Judge without whether a set of objects has less counting whether a set of than, more than, or the same objects has less than, more than, number of objects as a reference or the same number of objects set. as a reference set. 4.1.C.2 – Determine the Pg. 314, No. 6: Determine the reasonableness of an answer reasonableness of an answer by by estimating the result of estimating the result of computations (e.g., 15 + 16 is operations. not 211). 4.1.C.3 – Explore a variety of Pg. 313, No. 4: Explore, strategies for estimating both construct, and use a variety of quantities (e.g., the number of estimation strategies. 34 marbles in a jar) and results of Pg. 258, No. 5: Use a variety of computation. mental computation and estimation techniques. Pg. 259, No. 6: Select and use appropriate computational methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results. Pg. 288, No. 6: Understand and incorporate estimation and repeated measures in measurement activities. 4.2 All students will develop 4.2.A. Geometric Properties Pgs. 213-214, Overview spatial sense and the ability to use geometric properties, 4.2.A.1 – Identify and describe Pg. 215, No. 1: Explore spatial relationships, and spatial relationships among relationships such as the measurement to model, objects in space and their relative direction, orientation, and describe and analyze shapes and sizes. perspectives of objects in space, phenomena. • Inside/outside, left/right, their relative shapes and sizes, 4.2.A. Geometric Properties above/below, between and the relations between 4.2.B. Transforming Shapes • Smaller/larger/same size, objects and their shadows or 4.2.C. Coordinate Geometry wider/narrower, projections. 4.2.D. Units of Measurement longer/shorter 4.2.E. Measuring Geometric • Congruence (i.e. same Objects size and shape) 4.2.A.2 – Use concrete objects, Pg. 215, No. 3: Explore drawings, and computer graphics properties of three- and two- to identify, classify, and describe dimensional shapes using standard three-dimensional and concrete objects, drawings, and two-dimensional shapes. computer graphics. • Vertex, edge, face, side Pg. 216, No. 4: Use properties • 3D figures – cube, of three- and two-dimensional 35 rectangular prism, shapes to identify, classify, and sphere, cone, cylinder, describe shapes. and pyramid • 2D figures – square, rectangle, circle, triangle • Relationships between three- and two- dimensional shapes (i.e., the face of a 3D shape is a 2D shape) 4.2.A.3 – Describe, identify and Pg. 215, No. 2: Explore create instances of line relationships among shapes, symmetry. such as congruence, symmetry, similarity, and self-similarity. 4.2.A.4 – Recognize, describe, Pg. 217, No. 7: Explore extend and create designs and geometric transformations such patterns with geometric objects as rotations (turns), reflections of different shapes and colors. (flips), and translations (slides). Pg. 217, No. 9: Understand the variety of ways in which geometric shapes and objects can be measured. Pg. 217, No. 10: Investigate the occurrence of geometry in nature, art, and other areas. Pg. 216, No. 6: Use tessellations to explore properties of geometric shapes and their relationships to the concepts of area and perimeter. 36 4.2.B. Transforming Shapes Pgs. 338-339, Overview 4.2.B.1 – Use simple shapes to Pg. 340, No. 1: Reproduce, make designs, patterns and extend, create, and describe pictures. patterns and sequences using a variety of materials. Pg. 342, No. 5: Observe and recognize examples of patterns, relationships, and functions in other disciplines and contexts. Pg. 343, No. 6: Form and verify generalizations based on observations of patterns and relationships. Pg. 216, No. 6: Use tessellations to explore properties of geometric shapes and their relationships to the concepts of area and perimeter. Pg. 217, No. 9: Understand the variety of ways in which geometric shapes and objects can be measured. Pg. 217, No. 10: Investigate the occurrence of geometry in nature, art, and other areas. 4.2.B.2 – Combine and Pg. 216, No. 5: Investigate and subdivide simple shapes to predict the results of combining, make other shapes. subdividing, and changing shapes. 4.2.C. Coordinate Geometry Pgs. 445-447, Overview 37 4.2.C.1 – Give and follow Pg. 217, No. 8: Develop the directions for getting from one concepts coordinates and paths, point to another on a map or using maps, tables, and grids. grid. Pg. 450, No. 5: Follow, devise, and describe practical lists of instructions. 4.2.D. Units of Measurement Pgs. 284-285, Overview 4.2.D.1 – Directly compare and Pg. 287, No. 2: Compare and order objects according to order objects according to some measurable attributes. measurable attribute. • Attributes – length, Pg. 288, No. 4: Develop and weight, capacity, time, use personal referents for temperature. standard units of measure (such as the width of a finger to approximate a centimeter). 4.2.D.2 – Recognize the need for Pg. 288, No. 3: Recognize the a uniform unit of measurement. need for a uniform unit of measure. 4.2.D.3 – Select and use Pg. 288, No. 5: Select and use appropriate standard and non- appropriate standard and non- standard units of measure and standard units of measurement standard measurement tools to to solve real-life problems. solve real-life problems. • Length – inch, foot, yard, centimeter, meter • Weight – pound, gram, kilogram • Capacity – pint, quart, liter 38 • Time – second, minute, hour, day, week, month, year • Temperature – degrees Celsius, degrees Fahrenheit 4.2.D.4 – Estimate measures. Pg. 288, No. 6: Understand and incorporate estimation and repeated measures in measurement activities. 4.2.E. Measuring Geometric Pgs. 213-214, Overview Objects 4.2.E.1 – Directly measure the Pg. 215, No. 3: Explore perimeter of simple two- properties of three- and two- dimensional shapes. dimensional shapes using concrete objects, drawings, and computer graphics. Pg. 286, No. 1: Use and describe measures of length, distance, capacity, weight, area, volume, time, and temperature. 4.2.E.2 – Directly measure the Pg. 216, No. 6: Use area of simple two-dimensional tessellations to explore shapes by covering them with properties of geometric shapes squares. and their relationships to the concepts of area and perimeter. 4.3 All students will represent 4.3.A. Patterns Pgs. 338-339, Overview and analyze relationships 39 among variable quantities 4.3.A.1 – Recognize, describe, Pg. 340, No. 1: Reproduce, and solve problems extend, and create patterns. extend, create, and describe involving patterns, • Using concrete materials patterns and sequences using a functions, and algebraic (manipulatives), pictures, variety of materials. concepts and processes. rhythms, and whole Pg. 341, No. 2: Use tables, 4.3.A Patterns numbers. rules, variables, open sentences, 4.3.B Functions and • Descriptions using words and graphs to describe patterns Relationships and symbols (e.g., “add and other relationships. 4.3.C Modeling two” or “+2”) 4.3.D Procedures • Repeating patterns • Whole-number patterns that grow or shrink as a result of repeatedly adding or subtracting a fixed number (e.g., skip counting forward or backward) 4.3.B. Functions and Pgs. 338-339, Overview Relationships 4.3.B.1 – Use concrete and Pg. 341, No. 3: Use concrete pictorial models of function and pictorial models to explore machines to explore the basic the basic concept of a function. concept of a function. 4.3.C. Modeling Pg. 490, Overview 4.3.C.1 – Recognize and Pg. 491, No. 2: Investigate and describe changes over time (e.g., describe how certain quantities temperature, height). change over time. 4.3.C.2 – Construct and solve Pgs. 408-409, Overview simple open sentences Pg. 411, No. 4: Construct and 40 involving addition or solve open sentences (examples: subtraction. 3 + )7 = ٱthat describe real-life • Result unknown (e.g., 6 situations. – 2 = __ or n = 3 + 5) • Part unknown (e.g., 3 + __ = 8) 4.3.D. Procedures Pgs. 253-255, Overview 4.3.D.1 – Understand and Pg. 259, No. 7: Understand and apply (but don’t name) the use relationships among following properties of operations and properties of addition: operations. • Commutative (e.g., 5 + 3 = 3 + 5) • Zero as the identity element (e.g., 7 + 0 = 7) • Associative (e.g., 7.+ 3 + 2 can be found by first adding either 7 + 3 or 3 + 2) 4.4 All students will develop an 4.4.A. Data Analysis Pgs. 445-447, Overview understanding of the concepts and techniques of 4.4.A.1 – Collect, generate, Pg. 376, No. 1: Formulate and data analysis, probability, record, and organize data in solve problems that involve and discrete mathematics, response to questions, claims, collecting, organizing, and and will use them to model or curiosity. analyzing data. situations, solve problems, • Data collected from and analyze and draw students’ everyday appropriate inferences from experiences data. • Data generated from 4.4.A. Data Analysis chance devices, such as 4.4.B. Probability spinners and dice 4.4.C. Discrete Mathematics – 41 Systematic Listing and 4.4.A.2 – Read, interpret, Pg. 376, No. 1: Formulate and Counting construct, and analyze displays solve problems that involve 4.4.D. Discrete Mathematics – of data. collecting, organizing, and Vertex-Edge Graphs and • Pictures, tally chart, analyzing data. Algorithms pictograph, bar graph, Pg. 377, No. 3: Make Venn diagram inferences and formulate • Smallest to largest, hypotheses based on data. most frequent (mode) Pg. 377, No. 4: Understand and informally use the concepts of range, mean, mode, and median. Pg. 377, No. 5: Construct, read, and interpret displays of data such as pictographs, bar graphs, circle graphs, tables, and lists. 4.4.B. Probability Pgs. 374-375, Overview 4.4.B.1 – Use chance devices Pg. 376, No. 2: Generate and like spinners and dice to analyze data obtained using explore concepts of chance devices such as spinners probability. and dice. • Certain, impossible Pg. 378, No. 6: Determine the • More likely, less likely, probability of a simple event, equally likely assuming equally likely outcomes. 4.4.B.2 – Provide probability of Pg. 378, No. 6: Determine the specific outcomes. probability of a simple event, • Probability of getting assuming equally likely specific outcome when a outcomes. coin is tossed, when die is rolled, when spinner Pg. 378, No. 7: Make is spun (e.g., if spinner predictions that are based on has five equal sectors, intuitive, experimental, and 42 then probability of theoretical probabilities. getting a particular Pg. 378, No. 8: Use concepts of sector is one out of five) certainty, fairness, and chance • When picking a marble to discuss the probability of from a bag with three actual events. red marbles and four blue marbles, the probability of getting a red marble is three out of seven. 4.4.C. Discrete Mathematics – Pgs. 445-447, Overview Systematic Listing and Counting 4.4.C.1 – Sort and classify Pg. 449, No. 4: Investigate objects according to attributes. ways to represent and classify • Venn diagrams data according to attributes, such as shape or color, and relationships, and discuss the purpose and usefulness of such classification. 4.4.C.2 – Generate all Pg. 447, No. 1: Explore a possibilities in simple counting variety of puzzles, games, and situations (e.g., all outfits counting problems. involving two shirts and three pants). 4.4.D. Discrete Mathematics – Pgs. 445-447, Overview Vertex-Edge Graphs and Algorithms 43 4.4.D.1 – Follow simple sets of Pg. 450, No. 5: Follow, devise, directions (e.g., from one and describe practical lists of location to another, or from a instructions. recipe). Pg. 448, No. 2: Use networks and tree diagrams to represent everyday situations. 4.4.D.2 – Color simple maps Pg. 448, No. 2: Use networks with a small number of colors. and tree diagrams to represent everyday situations. 4.4.D.3 – Play simple two-person Pg. 447, No. 1: Explore a games (e.g., tic-tac-toe) and variety of puzzles, games, and informally explore the idea of counting problems. what the outcome should be. 44 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 45 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 46 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 47 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 48 MATHEMATICS SECOND GRADE STANDARD STUDENT OUTCOME SUGGESTED ACTIVITIES TEACHER’S NOTES AND NJ FRAMEWORKS 1996 SUPPLEMENTARY RESOURCES 4.1 All students will develop 4.1.A. Number Sense Pgs. 176-177, Overview number sense and will perform standard numerical 4.1.A.1 – Use real-life Pg. 178, No. 1: Use real-life operations and estimations experiences, physical materials, experiences, physical materials, on all types of numbers in a and technology to construct and technology to construct variety of ways. meanings for numbers. meanings for whole numbers, 4.1.A. Number Sense • Whole numbers through commonly used fractions, and 4.1.B. Numerical Operations hundreds decimals. 4.1.C. Estimation • Ordinals • Proper fractions (denominators of 2, 3, 4, 8, 10) 4.1.A.2 – Demonstrate an Pg. 179, No. 4: Develop a sense understanding of whole number of the magnitudes of whole place value concepts. numbers, commonly used fractions, and decimals. 4.1.A.3 – Understand that Pg. 180, No. 5: Understand the numbers have a variety of uses. various uses of numbers including counting, measuring, labeling, and indicating location. 4.1.A.4 – Count and perform Pg. 180, No. 6: Count and simple computations with coins. perform simple computations • Amounts up to $1.00 with money. (using cents notation) 49 4.1.A.5 – Compare and order Pg. 181, No. 8: Compare and whole numbers. order whole numbers, commonly used fractions, and decimals. 4.1.B. Numerical Operations Pgs. 253-255, Overview 4.1.B.1 – Develop the meanings Pg. 256, No. 1: Develop of addition and subtraction by meaning for the four basic concretely modeling and arithmetic operations by discussing a large variety of modeling and discussing a problems. variety of problems. • Joining, separating, and Pg. 257, No. 2: Develop comparing proficiency with and memorize basic number facts using a variety of fact strategies (such as “counting on” and “doubles”). Pg. 257, No. 3: Construct, use and explain procedures for performing whole number calculations in the various methods of computation. Pg. 258, No. 5: Use a variety of mental computation and estimation techniques. Pg. 259, No. 6: Select and use appropriate computational methods from mental math, estimation, paper-and-pencil, and calculator methods, and check, the reasonableness of results. Pg. 259, No. 7: Understand and use relationships among 50 operations and properties of operations. 4.1.B.2 – Explore the meanings Pg. 256, No. 1: Develop of multiplication and division meaning for the four basic by modeling and discussing arithmetic operations by problems. modeling and discussing a variety of problems. Pg. 259, No. 7: Understand and use relationships among operations and properties of operations. 4.1.B.3 – Develop proficiency Pg. 257, No. 2: Develop with basic addition and proficiency with and memorize subtraction facts using a variety basic number facts using a of fact strategies (such as variety of fact strategies (such as “counting on” and “near “counting on” and “doubles”). doubles”) and then commit them to memory. 4.1.B.4 – Construct, use, and Pg. 257, No. 3: Construct, use, explain procedures for and explain procedures for performing addition and performing whole number subtraction calculations with: calculations in the various • Pencil-and-paper methods of computation. • Mental math • Calculator 4.1.B.5 – Use efficient and Pg. 259, No. 6: Select and use accurate pencil-and-paper appropriate computational procedures for computation with methods from mental math, whole numbers. estimation, paper-and-pencil, • Addition of 2-digit and calculator methods, and numbers check the reasonableness of 51 • Subtraction of 2-digit results. numbers 4.1.B.6 – Select pencil-and- Pg. 259, No. 6: Select and use paper, mental math, or a appropriate computational calculator as the appropriate methods from mental math, computational method in a given estimation, paper-and-pencil, situation depending on the and calculator methods, and context and numbers. check the reasonableness of results. 4.1.B.7 – Check the Pg. 259, No. 6: Select and use reasonableness of results of appropriate computational computations. methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results. Pg. 258, No. 5: Use a variety of mental computation and estimation techniques. 4.1.B.8 – Understand and use the Pg. 259, No. 7: Understand and inverse relationship between use relationships among addition and subtraction. operations and properties of operations. 4.1.C. Estimation Pg. 311, Overview 4.1.C.1 – Judge without counting Pg. 312, No. 1: Judge without whether a set of objects has less counting whether a set of than, more than, or the same objects has less than, more than, number of objects as a reference or the same number of objects set. as a reference set. 52 4.1.C.2 – Determine the Pg. 314, No. 6: Determine the reasonableness of an answer by reasonableness of an answer by estimating the result of estimating the result of computations (e.g., 15 + 16 is operations. not 211). 4.1.C.3 – Explore a variety of Pg. 313, No. 4: Explore, strategies for estimating both construct, and use a variety of quantities (e.g., the number of estimation strategies. marbles in a jar) and results of Pg. 258, No. 5: Use a variety of computation. mental computation and estimation techniques. Pg. 259, No. 6: Select and use appropriate computational methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results. Pg. 288, No. 6: Understand and incorporate estimation and repeated measures in measurement activities. 4.2. All students will develop 4.2.A. Geometric Properties Pgs. 213-214, Overview spatial sense and the ability to use geometric properties, 4.2.A.1 – Identify and describe Pg. 215, No. 1: Explore spatial relationships, and spatial relationships among relationships such as the measurement to model, objects in space and their relative direction, orientation, and describe and analyze shapes and sizes. perspectives of objects in space, phenomena. • Inside/outside, left/right, their relative shapes and sizes, 4.2.A. Geometric Properties above/below, between and the relations between 4.2.B. Transforming Shapes • Smaller/larger/same size, objects and their shadows or 4.2.C. Coordinate Geometry wider/narrower, projections. 4.2.D. Units of Measurement longer/shorter 53 4.2.E. Measuring Geometric • Congruence (i.e. same Objects size and shape) 4.2.A.2 – Use concrete objects, Pg. 215, No. 3: Explore drawings, and computer graphics properties of three- and two- to identify, classify, and describe dimensional shapes using standard three-dimensional and concrete objects, drawings, and two-dimensional shapes. computer graphics. • Vertex, edge, face, side Pg. 216, No. 4: Use properties • 3D figures – cube, of three- and two-dimensional rectangular prism, sphere, shapes to identify, classify, and cone, cylinder, and describe shapes. pyramid • 2D figures – square, rectangle, circle, triangle • Relationships between three- and two- dimensional shapes (i.e., the face of a 3D shape is a 2D shape) 4.2.A.3 – Describe, identify and Pg. 215, No. 2: Explore create instances of line relationships among shapes, symmetry. such as congruence, symmetry, similarity, and self-similarity. 4.2.A.4 – Recognize, describe, Pg. 217, No. 7: Explore extend and create designs and geometric transformations such patterns with geometric objects as rotations (turns), reflections of different shapes and colors. (flips), and translations (slides). Pg. 217, No. 9: Understand the variety of ways in which geometric shapes and objects can be measured. 54 Pg. 217, No. 10: Investigate the occurrence of geometry in nature, art, and other areas. Pg. 216, No. 6: Use tessellations to explore properties of geometric shapes and their relationships to the concepts of area and perimeter. 4.2.B. Transforming Shapes Pgs. 338-339, Overview 4.2.B.1 – Use simple shapes to Pg. 340, No. 1: Reproduce, make designs, patterns and extend, create, and describe pictures. patterns and sequences using a variety of materials. Pg. 342, No. 5: Observe and recognize examples of patterns, relationships, and functions in other disciplines and contexts. Pg. 343, No. 6: Form and verify generalizations based on observations of patterns and relationships. Pg. 216, No. 6: Use tessellations to explore properties of geometric shapes and their relationships to the concepts of area and perimeter. Pg. 217, No. 9: Understand the variety of ways in which geometric shapes and objects can be measured. Pg. 217, No. 10: Investigate the occurrence of geometry in 55 nature, art, and other areas. 4.2.B.2 – Combine and Pg. 216, No. 5: Investigate and subdivide simple shapes to make predict the results of combining, other shapes. subdividing, and changing shapes. 4.2.C. Coordinate Geometry Pgs. 445-447, Overview 4.2.C.1 – Give and follow Pg. 217, No. 8: Develop the directions for getting from one concepts coordinates and paths, point to another on a map or using maps, tables, and grids. grid. Pg. 450, No. 5: Follow, devise, and describe practical lists of instructions. 4.2.D. Units of Measurement Pgs. 284-285, Overview 4.2.D.1 – Directly compare and Pg. 287, No. 2: Compare and order objects according to order objects according to some measurable attributes. measurable attribute. • Attributes – length, Pg. 288, No. 4: Develop and weight, capacity, time, use personal referents for temperature. standard units of measure (such as the width of a finger to approximate a centimeter). 4.2.D.2 – Recognize the need for Pg. 288, No. 3: Recognize the a uniform unit of measurement. need for a uniform unit of measure. 4.2.D.3 – Select and use Pg. 288, No. 5: Select and use appropriate standard and non- appropriate standard and non- 56 standard units of measure and standard units of measurement standard measurement tools to to solve real-life problems. solve real-life problems. • Length – inch, foot, yard, centimeter, meter • Weight – pound, gram, kilogram • Capacity – pint, quart, liter • Time – second, minute, hour, day, week, month, year • Temperature – degrees Celsius, degrees Fahrenheit 4.2.D.4 – Estimate measures. Pg. 288, No. 6: Understand and incorporate estimation and repeated measures in measurement activities. 4.2.E. Measuring Geometric Pgs. 213-214, Overview Objects 4.2.E.1 – Directly measure the Pg. 215, No. 3: Explore perimeter of simple two- properties of three- and two- dimensional shapes. dimensional shapes using concrete objects, drawings, and computer graphics. Pg. 286, No. 1: Use and describe measures of length, distance, capacity, weight, area, volume, time, and temperature. 57 4.2.E.2 – Directly measure the Pg. 216, No. 6: Use area of simple two-dimensional tessellations to explore shapes by covering them with properties of geometric shapes squares. and their relationships to the concepts of area and perimeter. 4.3 All students will represent 4.3.A. Patterns Pgs. 338-339, Overview and analyze relationships among variable quantities 4.3.A.1 – Recognize, describe, Pg. 340, No. 1: Reproduce, and solve problems extend, and create patterns. extend, create, and describe involving patterns, • Using concrete materials patterns and sequences using a functions, and algebraic (manipulatives), pictures, variety of materials. concepts and processes. rhythms, and whole Pg. 341, No. 2: Use tables, 4.3.A. Patterns numbers. rules, variables, open sentences, 4.3.B. Functions and • Descriptions using words and graphs to describe patterns Relationships and symbols (e.g., “add and other relationships. 4.3.C. Modeling two” or “+2”) 4.3.D. Procedures • Repeating patterns • Whole-number patterns that grow or shrink as a result of repeatedly adding or subtracting a fixed number (e.g., skip counting forward or backward) 4.3.B. Functions and Pgs. 338-339, Overview Relationships 4.3.B.1 – Use concrete and Pg. 341, No. 3: Use concrete pictorial models of function and pictorial models to explore machines to explore the basic the basic concept of a function. concept of a function. 58 4.3.C. Modeling Pg. 490, Overview 4.3.C.1 – Recognize and Pg. 491, No. 2: Investigate and describe changes over time (e.g., describe how certain quantities temperature, height). change over time. 4.3.C.2 – Construct and solve Pgs. 408-409, Overview simple open sentences involving Pg. 411, No. 4: Construct and addition or subtraction. solve open sentences (examples: • Result unknown (e.g., 6 – 3 + )7 = ٱthat describe real-life 2 = __ or n = 3 + 5) situations. • Part unknown (e.g., 3 + __ = 8) 4.3.D. Procedures Pgs. 253-255, Overview 4.3.D.1 – Understand and apply Pg. 259, No. 7: Understand and (but don’t name) the following use relationships among properties of addition: operations and properties of • Commutative (e.g., 5 + 3 operations. = 3 + 5) • Zero as the identity element (e.g., 7 + 0 = 7) • Associative (e.g., 7.+ 3 + 2 can be found by first adding either 7 + 3 or 3 + 2) 4.4 All students will develop an 4.4.A. Data Analysis Pgs. 445-447, Overview understanding of the concepts and techniques of 4.4.A.1 – Collect, generate, Pg. 376, No. 1: Formulate and data analysis, probability, record, and organize data in solve problems that involve and discrete mathematics, response to questions, claims, or collecting, organizing, and and will use them to model curiosity. analyzing data. situations, solve problems, • Data collected from 59 and analyze and draw students’ everyday appropriate inferences from experiences data. • Data generated from 4.4.A. Data Analysis chance devices, such as 4.4.B. Probability spinners and dice 4.4.C. Discrete Mathematics – Systematic Listing and 4.4.A.2 – Read, interpret, Pg. 376, No. 1: Formulate and Counting construct, and analyze displays solve problems that involve 4.4.D. Discrete Mathematics – of data. collecting, organizing, and Vertex-Edge Graphs and • Pictures, tally chart, analyzing data. Algorithms pictograph, bar graph, Pg. 377, No. 3: Make Venn diagram inferences and formulate • Smallest to largest, most hypotheses based on data. frequent (mode) Pg. 377, No. 4: Understand and informally use the concepts of range, mean, mode, and median. Pg. 377, No. 5: Construct, read, and interpret displays of data such as pictographs, bar graphs, circle graphs, tables, and lists. 4.4.B. Probability Pgs. 374-375, Overview 4.4.B.1 – Use chance devices Pg. 376, No. 2: Generate and like spinners and dice to explore analyze data obtained using concepts of probability. chance devices such as spinners • Certain, impossible and dice. • More likely, less likely, Pg. 378, No. 6: Determine the equally likely probability of a simple event, assuming equally likely outcomes. 4.4.B.2 – Provide probability of Pg. 378, No. 6: Determine the 60 specific outcomes. probability of a simple event, • Probability of getting assuming equally likely specific outcome when a outcomes. coin is tossed, when die Pg. 378, No. 7: Make is rolled, when spinner is predictions that are based on spun (e.g., if spinner has intuitive, experimental, and five equal sectors, then theoretical probabilities. probability of getting a Pg. 378, No. 8: Use concepts of particular sector is one certainty, fairness, and chance to out of five) discuss the probability of actual • When picking a marble events. from a bag with three red marbles and four blue marbles, the probability of getting a red marble is three out of seven. 4.4.C. Discrete Mathematics – Pgs. 445-447, Overview Systematic Listing and Counting 4.4.C.1 – Sort and classify Pg. 449, No. 4: Investigate objects according to attributes. ways to represent and classify • Venn diagrams data according to attributes, such as shape or color, and relationships, and discuss the purpose and usefulness of such classification. 4.4.C.2 – Generate all Pg. 447, No. 1: Explore a possibilities in simple counting variety of puzzles, games, and situations (e.g., all outfits counting problems. involving two shirts and three pants). 61 4.4.D. Discrete Mathematics – Pgs. 445-447, Overview Vertex-Edge Graphs and Algorithms 4.4.D.1 – Follow simple sets of Pg. 450, No. 5: Follow, devise, directions (e.g., from one and describe practical lists of location to another, or from a instructions. recipe). Pg. 448, No. 2: Use networks and tree diagrams to represent everyday situations. 4.4.D.2 – Color simple maps Pg. 448, No. 2: Use networks with a small number of colors. and tree diagrams to represent everyday situations. 4.4.D.3 – Play simple two- Pg. 447, No. 1: Explore a person games (e.g., tic-tac-toe) variety of puzzles, games, and and informally explore the idea counting problems. of what the outcome should be. Pg. 448, No. 2: Use networks 4.4.D.4 – Explore concrete and tree diagrams to represent models of vertex-edge graphs everyday situations. (e.g., vertices as “islands” and edges as “bridges”). • Paths from one vertex to another 62 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 63 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 64 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 65 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 66 MATHEMATICS GRADE 3 STANDARD STUDENT OUTCOME SUGGESTED ACTIVITIES TEACHER’S NOTES AND NJ FRAMEWORKS 1996 SUPPLEMENTARY RESOURCES 4.5 All students will develop 4.1.A. Number Sense Pgs. 183-184, Overview number sense and will perform standard numerical 4.1.A.1 – Use real-life Pg. 185, No. 1: Use real-life operations and estimations experiences, physical materials, experiences, physical materials, on all types of numbers in a and technology to construct and technology to construct variety of ways. meanings for numbers (unless meanings for whole numbers, 4.1.A. Number Sense otherwise noted, all indicators commonly used fractions, and 4.1.B. Numerical Operations for grade 3 pertain to these sets decimals. 4.1.C. Estimation of numbers as well). Pg. 186, No. 4: Develop a sense • Whole numbers through of the magnitudes of whole hundred thousands numbers, commonly used • Commonly used fractions fractions, and decimals. (denominators of 2, 3, 4, Pg. 187, No. 7: Use models to 5, 6, 8, 10) as part of a relate whole numbers, commonly whole, as a subset of a used fractions, and decimals to set, and as a location on a each other, and to represent number line equivalent forms of the same number. 4.1.A.2 – Demonstrate an Pg. 185, No. 2: Develop an understanding of whole number understanding of place value place value concepts concepts and numeration in • Standard, expanded, and relationship to counting and written form. grouping. 67 4.1.A.3 – Identify whether any whole number is odd or even. 4.1.A.4 – Explore the extension Pg. 186, No. 4: Develop a sense of the place value system to of the magnitudes of whole decimals through hundredths. numbers, commonly used fractions, and decimals. Pg. 187, No. 7: Use models to relate whole numbers, commonly used fractions, and decimals to each other, and to represent equivalent forms of the same number. 4.1.A.5 – Understand the various Pg. 187, No. 5: Understand the uses of numbers. various uses of numbers • Counting, measuring, including counting, measuring, labeling (e.g., numbers on labeling, and indicating location. baseball uniforms) • Ordinal numbers 4.1.A.6 – Compare and order Pg. 188, No. 8: Compare and numbers. order whole numbers, commonly used fractions, and decimals. 4.1.B. Numerical Operations Pgs. 261-262, Overview 4.1.B.1 – Develop the meanings Pg. 263, No. 1: Develop of the four basic arithmetic meaning for the four basic operations by modeling and arithmetic operations by discussing a large variety of modeling and discussing a problems. variety of problems. 68 • Addition and subtraction: joining, separating, comparing • Multiplication: repeated addition, area/array • Division: repeated subtraction, sharing • Zero as place holder 4.1.B.2 – Develop proficiency Pg. 263, No. 2: Develop with basic multiplication and proficiency with and memorize division number facts using a basic number facts using a variety of fact strategies (such as variety of fact strategies (such as “skip counting” and “repeated “counting on” and “doubles”). subtraction”). 4.1.B.3 – Construct, use, and Pg. 264, No. 3: Construct, use, explain procedures for and explain procedures for performing whole number performing whole number calculations with: calculations in the various • Pencil-and-paper methods of computation. • Mental math Pg. 266, No. 6: Select and use appropriate computational • Calculator methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results. 69 4.1.B.4 – Use efficient and Pg. 266, No. 6: Select and use accurate pencil-and-paper appropriate computational procedures for computation with methods from mental math, whole numbers. estimation, paper-and-pencil, and • Addition of 3-digit calculator methods, and check numbers the reasonableness of results. • Subtraction of 3-digit numbers • Multiplication of 2-digit numbers by 1-digit numbers 4.1.B.5 – Count and perform Pg. 187, No. 6: Count and simple computations with money. perform simple computations • Cents notation (¢) with money • Understand relationship between penny, nickel, dime, quarter, half dollar, and dollar. 4.1.B.6 – Select pencil-and- Pg. 265, No. 5: Use a variety of paper, mental math, or a mental computation and calculator as the appropriate estimation techniques. computational method in a given situation depending on the context and numbers. 4.1.B.7 – Check the Pg. 319, No. 6: Determine the reasonableness of results of reasonableness of an answer by computations. estimating the result of operations. 70 4.1.C. Estimation Pg. 316, Overview 4.1.C.1 – Judge without counting Pg. 317, No. 1: Judge without whether a set of objects has less counting whether a set of objects than, more than, or the same has less than, more than, or the number of objects as a reference same number of objects as a set. reference set. 4.1.C.2 – Construct and use a Pg. 265, No. 5: Use a variety of variety of estimation strategies mental computation and (e.g., rounding and mental math) estimation techniques. for estimating both quantities and Pg. 318, No. 4: Explore, the results of computations. construct, and use a variety of estimation strategies. Pg. 319, No. 7: Apply estimation in working with quantities, measurement, time, computation, and problem solving. 4.1.C.3 – Recognize when an Pg. 318, No. 5: Recognize when estimate is appropriate, and estimation is appropriate, and understand the usefulness of an understand the usefulness of an estimate as distinct from an exact estimate as distinct from an exact answer. answer. 4.1.C.4 – Use estimation to Pg. 319, No. 6: Determine the determine whether the result of a reasonableness of an answer by computation (either by calculator estimating the result of or by hand) is reasonable. operations. 71 4.6 All students will develop 4.2.A. Geometric Properties Pgs. 219-220, Overview spatial sense and the ability to use geometric properties, 4.2.A.1 – Identify and describe Pg. 221, No. 1: Explore spatial relationships, and spatial relationships of two or relationships such as the measurement to model, more objects in space. direction, orientation, and describe and analyze • Direction, orientation, and perspectives of objects in space, phenomena. perspectives (e.g., which their relative shapes and sizes, 4.2.A. Geometric Properties object is on your left and the relations between objects 4.2.B. Transforming Shapes when you are standing and their shadows or projections. 4.2.C. Coordinate Geometry here?) 4.2.D. Units of Measurement • Relative shapes and sizes 4.2.E. Measuring Geometric Objects 4.2.A.2 – Use properties of Pg. 222, No. 3: Explore standard three-dimensional and properties of three- and two- two-dimensional shapes to dimensional shapes using identify, classify, and describe concrete objects, drawings, and them. computer graphics. • Vertex, edge, face, side, Pg. 222, No. 4: Use properties of angle three- and two-dimensional • 3D figures – cube, shapes to identify , classify, and rectangular prism, sphere, describe shapes. cone, cylinder, and Pg. 222, No. 5: Investigate and pyramid predict the results of combining, • 2D figures – square, subdividing, and changing rectangle, circle, triangle, shapes. pentagon, hexagon, octagon 4.2.A.3 – Identify and describe Pg. 221, No. 2: Explore relationships among two- relationships among shapes, such dimensional shapes. as congruence, symmetry, similarity, and self-similarity. 72 • Same size, same shape, congruency • Lines of symmetry 4.2.A.4 – Understand and apply concepts involving lines, angles, and circles. • Line, line segment, endpoint 4.2.A.5 – Recognize, describe, extend and create space-filling patterns. 4.2.B. Transforming Shapes 4.2.B.1 – Describe and use Pg. 223, No. 7: Explore geometric transformations (slide, geometric transformations such flip, turn). as rotations (turns), reflections (flips), and translations (slides). 4.2.B.2 – Investigate the Pg. 223, No. 10: Investigate the occurrence of geometry in nature occurrence of geometry in nature, and art. art, and other areas. Pg. 457, No. 3: Identify and investigate sequences and patterns found in nature, art, and music. 73 4.2.C. Coordinate Geometry 4.2.C.1 – Locate and name points Pg. 223, No. 8: Develop the in the first quadrant on a concepts of coordinates and coordinate grid. paths, using maps, tables, and grids. 4.2.D. Units of Measurement Pg. 290, Overview Pg. 493, Overview 4.2.D.1 – Understand that Pg. 223, No. 9: Understand the everyday objects have a variety variety of ways in which of attributes, each of which can geometric shapes and objects can be measured in many ways. be measured. 4.2.D.2 – Select and use Pg. 292, No. 2: Compare and appropriate standard units of order objects according to some measure and measurement tools measurable attribute. to solve real-life problems. Pg. 292, No. 5: Select and use • Length – fractions of an appropriate standard and non- inch (1/4, 1/2), foot, mile, standard units of measurement to decimeter, kilometer, solve real-life problems. meter Pg. 495, No. 3: Experiment with • Area – square inch, approximating length, area, and square centimeter volume, using informal • Weight – ounce, pound, measurement instruments. gram, kilogram Pg. 291, No. 1: Use and describe • Capacity – fluid ounce, measures of length, distance, cup, pint, quart, gallon, capacity, weight, area, volume, millimeter, liter time, and temperature. 74 4.2.D.3 – Incorporate estimation Pg. 317, No. 3: Visually estimate in measurement activities (e.g., length, area, volume, or angle estimate before measuring). measure. Pg. 293, No. 6: Understand and incorporate estimation and repeated measures in measurement activities. 4.2.E. Measuring Geometric Pg. 290, Overview Objects 4.2.E.1 – Determine the area of Pg. 291, No. 1: Use and describe simple two-dimensional shapes measures of length, distance, on a square grid. capacity, weight, area, volume, time, and temperature. 4.2.E.2 – Determine the Pg. 291, No. 1: Use and describe perimeter of simple shapes by measures of length, distance, measuring all of the sides. capacity, weight, area, volume, time, and temperature. 4.2.E.3 – Measure and compare Pg. 291, No. 1: Use and describe the volume of three-dimensional measures of length, distance, objects using materials such as capacity, weight, area, volume, rice or cubes. time, and temperature. 4.7 All students will represent 4.3.A. Patterns Pg. 345, Overview and analyze relationships among variable quantities 4.3.A.1 – Recognize, describe, Pg. 346, No. 1: Reproduce, and solve problems extend, and create patterns. extend, create, and describe involving patterns, functions, • Descriptions using words patterns and sequences using a and algebraic concepts and and number variety of materials. processes. sentences/expressions 75 4.3.A. Patterns • Whole number patterns Pg. 494, No. 2: Investigate and 4.3.B. Functions and that grow or shrink as a describe how certain quantities Relationships result of repeatedly change over time. 4.3.C. Modeling adding, subtracting, 4.3.D. Procedures multiplying by, or dividing by a fixed number (e.g., 5, 8,11 …or 800, 400, 200, …) 4.3.B. Functions and Pgs. 413-414, Overview Relationships 4.3.B.1 – Use concrete and Pg. 347, No. 2: Use tables, rules, pictorial models to explore the variables, open sentences, and basic concept of a function. graphs to describe patterns and • Input/output tables, T- other relationships. charts Pg. 347, No. 3: Use concrete and pictorial models to explore the basic concept of a function. 4.3.C. Modeling 4.3.C.1 – Recognize and describe Pg. 348, No. 4: Observe and change in quantities. explain how a change in one • Graphs representing physical quantity can produce a change over time (e.g., corresponding change in another. temperature, height) 76 4.3.C.2 – Construct and solve Pg. 415, No. 1: Understand and simple open sentences involving represent numerical situations addition or subtraction (e.g., 3 + using variables, expressions, and 6 = ___, n = 15 – 3, 3 + ___ = 3, number sentences. 16 – c = 7). Pg. 417, No. 4: Construct and solve open sentences (example: 3 + )7 = ٱthat describe real-life situations. Pg. 348, No. 5: Observe and recognize examples of patterns, relationships, and functions in other disciplines and contexts. Pg. 415, No. 2: Represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and number sentences, and translate from one to another. Pgs. 261-262, Overview 4.3.D. Procedures Pg. 416, No. 3: Understand and 4.3.D.1 – Understand and apply use properties of operations and the properties of operations and numbers. numbers. Pg. 266, No. 7: Understand and • Commutative (e.g., 3 X 7 use relationships among = 7 X 3) operations and properties of • Identify element for operations. multiplication is 1 (e.g., 1 X 8 = 8) 77 • Any number multiplied by zero is zero 4.3.D.2 – Understand and use the concepts of equals, less than, and greater than to describe relations between numbers. • Symbols (=, <, >) 4.8 All students will develop an 4.4.A. Data Analysis Pgs. 380-381, Overview understanding of the concepts and techniques of 4.4.A.1 – Collect, generate, Pg. 382, No. 1: Formulate and data analysis, probability, organize, and display data in solve problems that involve and discrete mathematics, response to questions, claims, or collecting, organizing, and and will use them to model curiosity. analyzing data. situations, solve problems, • Data collected from the and analyze and draw classroom environment appropriate inferences from data. 4.4.A.2 – Read, interpret, Pg. 383, No. 5: Construct, read, 4.4.A. Data Analysis construct, analyze, generate and interpret displays of data 4.4.B. Probability questions about, and draw such as pictographs, bar graphs, 4.4.C. Discrete Mathematics – inferences from displays of data. circle graphs, tables, and lists. Systematic Listing and Counting • Pictograph, bar graph, 4.4.D. Discrete Mathematics – table Vertex-Edge Graphs and Algorithms 4.4.B. Probability 4.4.B.1 – Use everyday events Pg. 382, No. 2: Generate and and chance devices, such as dice, analyze data obtained using coins, and unevenly divided chance devices such as spinners spinners, to explore concepts of and dice. 78 probability. Pg. 384, No. 6: Determine the • Likely, unlikely, certain, probability of a simple event impossible assuming equally likely • More likely, less likely, outcomes. equally likely 4.4.B.2 – Predict probabilities in Pg. 384, No. 7: Make predictions a variety of situations (e.g., given that are based on intuitive, the number of items of each color experimental, and theoretical in a bag, what is the probability probabilities. that an item picked will have a particular color). • What students think will happen (intuitive) • Collect data and use that data to predict the probability (experimental) 4.4.C. Discrete Mathematics – Pgs. 452-453, Overview Systematic Listing and Counting 4.4.C.1 – Represent and classify Pg. 458, No. 4: Investigate ways data according to attributes, such to represent and classify data as shape or color, and according to attributes, such as relationships. shape or color, and relationships, • Venn diagrams and discuss the purpose and • Numerical and usefulness of such classification. alphabetical order 79 4.4.C.2 – Represent all Pg. 454, No. 1: Explore a variety possibilities for a simple counting of puzzles, games, and counting situation in an organized way and problems. draw conclusions from this representation. • Organized lists, charts 4.4.D. Discrete Mathematics – Vertex-Edge Graphs and Algorithms 4.4.D.1 – Follow, devise, and Pg. 458, No. 5: Follow, devise, describe practical sets of and describe practical lists of directions (e.g., to add two 2- instructions. digit numbers). 4.4.D.2 – Explore vertex-edge graphs • Vertex, edge • Path 4.4.D.3 – Find the smallest Pg. 456, No. 2: Use networks number of colors needed to color and tree diagrams to represent a map. everyday situations. 80 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 81 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 82 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 83 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 84 MATHEMATICS GRADE 4 STANDARD STUDENT OUTCOME SUGGESTED ACTIVITIES TEACHER’S NOTES AND NJ FRAMEWORKS 1996 SUPPLEMENTARY RESOURCES 4.1 All students will develop 4.1.A. Number Sense Pgs. 183-184, Overview number sense and will perform standard numerical 4.1.A.1 – Use real-life Pg. 185, No. 1: Use real-life operations and estimations on experiences, physical materials, experiences, physical materials, all types of numbers in a and technology to construct and technology to construct variety of ways. meanings for numbers (unless meanings for whole numbers, 4.1.A. Number Sense otherwise noted, all indicators commonly used fractions, and 4.1.B. Numerical Operations for grade 4 pertain to these sets decimals. 4.1.C. Estimation of numbers as well). Pg. 186, No. 4: Develop a sense • Whole numbers through of the magnitudes of whole millions numbers, commonly used • Apply knowledge of fractions, and decimals. odd/evens • Commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 16) as part of a whole, as a subset of a set, and as a location on a number line • Read and write mixed numbers • Decimals through hundredths, i.e., money 85 4.1.A.2 – Demonstrate an Pg. 185, No. 2: Develop an understanding of place value understanding of place value concepts. concepts and numeration in relationship to counting and grouping. 4.1.A.3 – Demonstrate a sense of the relative magnitudes of numbers • Standard, expanded, and written form. 4.1.A.4 – Understand the various Pg. 187, No. 5: Understand the uses of numbers. various uses of numbers • Counting, measuring, including counting, measuring, labeling (e.g., numbers on labeling, and indicating location. baseball uniforms), locating (e.g., Room 235 is on the second floor) 4.1.A.5 – Use concrete and Pg. 187, No. 7: Use models to pictorial models to relate whole relate whole numbers, commonly numbers, commonly used used fractions, and decimals to fractions, and decimals to each each other, and to represent other, and to represent equivalent equivalent forms of the same forms of the same number. number. 4.1.A.6 – Compare and order Pg. 188, No. 8: Compare and numbers. order whole numbers, commonly used fractions, and decimals. 86 4.1.A.7 – Explore settings that Pg. 188, No. 9: Explore real-life give rise to negative numbers. settings which give rise to • Temperatures below 0º, negative numbers. debts • Extension of the number line 4.1.B. Numerical Operations Pgs. 261-262, Overview 4.1.B.1 – Develop the meanings Pg. 263, No. 1: Develop of the four basic arithmetic meaning for the four basic operations by modeling and arithmetic operations by discussing a large variety of modeling and discussing a problems. variety of problems. • Addition and subtraction: joining, separating, comparing • Multiplication: repeated addition, area/array • Division: repeated subtraction, sharing 4.1.B.2 – Develop proficiency Pg. 263, No. 2: Develop with basic multiplication and proficiency with and memorize division number facts using a basic number facts using a variety of fact strategies (such as variety of fact strategies (such as “skip counting” and “repeated “counting on” and “doubles”). subtraction”) and then commit them to memory. 87 4.1.B.3 – Construct, use, and Pg. 264, No. 3: Construct, use, explain procedures for and explain procedures for performing whole number performing whole number calculations with: calculations in the various • Pencil-and-paper methods of computation. • Mental math • Calculator 4.1.B.4 – Use efficient and Pg. 266, No. 6: Select and use accurate pencil-and-paper appropriate computational procedures for computation with methods from mental math, whole numbers. estimation, paper-and-pencil, and • Addition of 3-digit calculator methods, and check numbers the reasonableness of results. • Subtraction of 3-digit numbers • Multiplication of 2-digit numbers • Division of 3-digit numbers by 1-digit numbers • Zero as place holder 4.1.B.5 – Construct and use procedures for performing decimal addition and subtraction. 4.1.B.6 – Count and perform Pg. 187, No. 6: Count and simple computations with money. perform simple computations • Standard dollars and cents with money notation 88 4.1.B.7 – Select pencil-and- Pg. 265, No. 5: Use a variety of paper, mental math, or a mental computation and calculator as the appropriate estimation techniques. computational method in a given Pg. 266, No. 6: Select and use situation depending on the appropriate computational context and numbers. methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results. 4.1.B.8 – Check the Pg. 319, No. 6: Determine the reasonableness of results of reasonableness of an answer by computations. estimating the result of operations. 4.1.B.9 – Use concrete models to Pg. 265, No. 4: Use models to explore addition and subtraction explore operations with fractions with fractions. and decimals. 4.1.B.10 – Understand and use Pg. 266, No. 7: Understand and the inverse relationships between use relationships among addition and subtraction and operations and properties of between multiplication and operations. division. 4.1.C. Estimation Pg. 316, Overview 4.1.C.1 – Judge without counting Pg. 317, No. 1: Judge without whether a set of objects has less counting whether a set of objects than, more than, or the same has less than, more than, or the number of objects as a reference same number of objects as a set. reference set. 89 4.1.C.2 – Construct and use a Pg. 265, No. 4: Use models to variety of estimation strategies explore operations with fractions (e.g., rounding and mental math) and decimals. for estimating both quantities and Pg. 265, No. 5: Use a variety of the results of computations. mental computation and estimation techniques. Pg. 319, No. 7: Apply estimation in working with quantities, measurement, time, computation, and problem solving. 4.1.C.3 – Recognize when an Pg. 318, No. 5: Recognize when estimate is appropriate, and estimation is appropriate, and understand the usefulness of an understand the usefulness of an estimate as distinct from an exact estimate as distinct from an exact answer. answer. 4.1.C.4 – Use estimation to Pg. 319, No. 6: Determine the determine whether the result of a reasonableness of an answer by computation (either by calculator estimating the result of or by hand) is reasonable. operations. 4.2 All students will develop 4.2.A. Geometric Properties Pgs. 219-220, Overview spatial sense and the ability to use geometric properties, 4.2.A.1 – Identify and describe Pg. 221, No. 1: Explore spatial relationships, and spatial relationships of two or relationships such as the measurement to model, more objects in space. direction, orientation, and describe and analyze • Direction, orientation, and perspectives of objects in space, phenomena. perspectives (e.g., which their relative shapes and sizes, 4.2.A. Geometric Properties object is on your left and the relations between objects 4.2.B. Transforming Shapes when you are standing and their shadows or projections. 4.2.C. Coordinate Geometry here?) 4.2.D. Units of Measurement 90 4.2.E. Measuring Geometric Objects • Relative shapes and sizes • Shadows (projections) of everyday objects 4.2.A.2 – Use properties of Pg. 222, No. 3: Explore standard three-dimensional and properties of three- and two- two-dimensional shapes to dimensional shapes using identify, classify, and describe concrete objects, drawings, and them. computer graphics. • Vertex, edge, face, side, Pg. 222, No. 4: Use properties of angle three- and two-dimensional • 3D figures – cube, shapes to identify, classify, and rectangular prism, sphere, describe shapes. cone, cylinder, and Pg. 222, No. 5: Investigate and pyramid predict the results of combining, • 2D figures – square, subdividing, and changing rectangle, circle, triangle, shapes. quadrilateral, pentagon, hexagon, octagon • Inclusive relationships – squares are rectangles, cubes are rectangular prisms 4.2.A.3 – Identify and describe Pg. 221, No. 2: Explore relationships among two- relationships among shapes, such dimensional shapes. as congruence, symmetry, similarity, and self-similarity. • Congruence • Lines of symmetry • Similarity 91 4.2.A.4 – Understand and apply concepts involving lines, angles, and circles. • Point, line, line segment, endpoint • Parallel, Perpendicular • Angles – acute, right, obtuse • Circles – diameter, radius, center 4.2.A.5 – Recognize, describe, extend and create space-filling patterns. 4.2.B. Transforming Shapes 4.2.B.1 – Use simple shapes to Pg. 222, No. 6: Use tessellations cover an area (tessellations). to explore properties of geometric shapes and their relationships to the concepts of area and perimeter. 4.2.B.2 – Describe and use Pg. 223, No. 7: Explore geometric transformations (slide, geometric transformations such flip, turn). as rotations (turns), reflections (flips), and translations (slides). 4.2.B.3 – Investigate the Pg. 223, No. 10: Investigate the occurrence of geometry in nature occurrence of geometry in nature, and art. art, and other areas. Pg. 457, No. 3: Identify and investigate sequences and 92 patterns found in nature, art, and music. 4.2.C. Coordinate Geometry 4.2.C.1 – Locate and name points Pg. 223, No. 8: Develop the in the first quadrant on a concepts of coordinates and coordinate grid. paths, using maps, tables, and grids. 4.2.C.2 – Use coordinates to give Pg. 223, No. 8: Develop the or follow directions from one concepts of coordinates and point to another on a map or grid. paths, using maps, tables, and grids. 4.2.D. Units of Measurement Pg. 290, Overview Pg. 493, Overview 4.2.D.1 – Understand that everyday objects have a variety of attributes, each of which can be measured in many ways. 4.2.D.2 – Select and use Pg. 495, No. 3: Experiment with appropriate standard units of approximating length, area, and measure and measurement tools volume, using informal to solve real-life problems. measurement instruments. • Length – fractions of an Pg. 291, No. 1: Use and describe inch (1/8, 1/4, 1/2), mile, measures of length, distance, decimeter, kilometer capacity, weight, area, volume, time, and temperature. 93 • Area – square inch, Pg. 292, No. 2: Compare and square centimeter order objects according to some • Volume – cubic inch, measurable attribute. cubic centimeter Pg. 292, No. 5: Select and use • Weight – ounce appropriate standard and non- • Capacity – fluid ounce, standard units of measurement to cup, gallon, millimeter solve real-life problems. 4.2.D.3 – Develop and use Pg. 292, No. 4: Develop and use personal referents to approximate personal referents for standard standard units of measure (e.g., a units of measure (such as the common paper clip is about an width of a finger to approximate inch long). a centimeter). Pg. 317, No. 2: Visually estimate length, area, volume, or angle measure. Pg. 292, No. 3: Recognize the need for a uniform unit of measure. 4.2.D.4 – Incorporate estimation Pg. 317, No. 3: Visually estimate in measurement activities (e.g., length, area, volume, or angle estimate before measuring). measure. Pg. 293, No. 6: Understand and incorporate estimation and repeated measures in measurement activities. 4.2.D.5 – Solve problems involving elapsed time. 94 4.2.E. Measuring Geometric Pg. 290, Overview Objects 4.2.E.1 – Determine the area of Pg. 291, No. 1: Use and describe simple two-dimensional shapes measures of length, distance, on a square grid. capacity, weight, area, volume, time, and temperature. 4.2.E.2 – Distinguish between Pg. 222, No. 6: Use tessellations perimeter and area and use each to explore properties of appropriately in problem-solving geometric shapes and their situations. relationships to the concepts of area and perimeter. Pg. 291, No. 1: Use and describe measures of length, distance, capacity, weight, area, volume, time, and temperature. 4.2.E.3 – Measure and compare Pg. 291, No. 1: Use and describe the volume of three-dimensional measures of length, distance, objects using materials such as capacity, weight, area, volume, rice or cubes. time, and temperature. 4.3 All students will represent 4.3.A. Patterns Pg. 345, Overview and analyze relationships among variable quantities and solve problems involving 4.3.A.1 – Recognize, describe, Pg. 346, No. 1: Reproduce, patterns, functions, and extend, and create patterns. extend, create, and describe 95 algebraic concepts and • Descriptions using words patterns and sequences using a processes. and number variety of materials. 4.3.A. Patterns sentences/expressions, Pg. 494, No. 1: Investigate and 4.3.B. Functions and graphs, tables, variables describe patterns that continue Relationships (e.g., shape, blank, or indefinitely. 4.3.C. Modeling letter) Pg. 494, No. 2: Investigate and 4.3.D. Procedures • Sequences that stop or describe how certain quantities that continue infinitely change over time. • Whole number patterns Pg. 349, No. 6: Form and verify that grow or shrink as a generalizations based on result of repeatedly observations of patterns and adding, subtracting, relationships. multiplying by, or dividing by a fixed number (e.g., 5, 8,11 …or 800, 400, 200, …) • Sequences can often be extended in more than one way (e.g., the next term after 1, 2, 4, … could be 8, or 7, or …) • Form and verify generalizations based on observations of patterns and relationships. 4.3.B. Functions and Pgs. 413-414, Overview Relationships 4.3.B.1 – Use concrete and Pg. 347, No. 2: Use tables, rules, pictorial models to explore the variables, open sentences, and basic concept of a function. graphs to describe patterns and other relationships. 96 • Input/output tables, T- Pg. 347, No. 3: Use concrete and charts pictorial models to explore the • Combining two functions basic concept of a function. machines • Reversing a function machine 4.3.C. Modeling 4.3.C.1 – Recognize and describe Pg. 348, No. 4: Observe and change in quantities. explain how a change in one • Graphs representing physical quantity can produce a change over time (e.g., corresponding change in another. temperature, height) • How change in one physical quantity can produce a corresponding change in another (e.g., pitch of a sound depends on the rate of vibration) 4.3.C.2 – Construct and solve Pg. 415, No. 1: Understand and simple open sentences involving represent numerical situations any one operation (e.g., 3 X 6 = using variables, expressions, and ___, n = 15 ÷ 3, 3 X ___ = 0, 16 number sentences. – c = 7) Pg. 417, No. 4: Construct and solve open sentences (example: 3 + )7 = ٱthat describe real-life situations. 97 Pg. 348, No. 5: Observe and recognize examples of patterns, relationships, and functions in other disciplines and contexts. 4.3.D. Procedures Pgs. 261-262, Overview 4.3.D.1 – Understand, name, and Pg. 416, No. 3: Understand and apply the properties of operations use properties of operations and and numbers. numbers. • Commutative (e.g., 3 X 7 Pg. 266, No. 7: Understand and = 7 X 3) use relationships among • Identify element for operations and properties of multiplication is 1 (e.g., 1 operations. X 8 = 8) • Associative (e.g., 2 X 4 X 25 can be found by first multiplying either 2 x 4 or 4 x 25) • Division by zero is undefined • Any number multiplied by zero is zero 4.3.D.2 – Understand and use the concepts of equals, less than, and greater than in simple number sentences. • Symbols (=, <, >) 98 4.4 All students will develop an 4.4.A. Data Analysis Pgs. 380-381, Overview understanding of the concepts and techniques of data 4.4.A.1 – Collect, generate, Pg. 382, No. 1: Formulate and analysis, probability, and organize, and display data in solve problems that involve discrete mathematics, and response to questions, claims, or collecting, organizing, and will use them to model curiosity. analyzing data. situations, solve problems, • Data collected from the and analyze and draw school environment appropriate inferences from data. 4.4.A.2 – Read, interpret, Pg. 383, No. 5: Construct, read, 4.4.A. Data Analysis construct, analyze, generate and interpret displays of data 4.4.B. Probability questions about, and draw such as pictographs, bar graphs, 4.4.C. Discrete Mathematics – inferences from displays of data. circle graphs, tables, and lists. Systematic Listing and Counting • Pictograph, bar graph, 4.4.D. Discrete Mathematics – line plot, line graph, table Vertex-Edge Graphs and • Average (mean), most Algorithms frequent (mode), middle term (median) 4.4.B. Probability Pgs. 380-381, Overview 4.4.B.1 – Use everyday events Pg. 382, No. 2: Generate and and chance devices, such as dice, analyze data obtained using coins, and unevenly divided chance devices such as spinners spinners, to explore concepts of and dice. probability. Pg. 384, No. 8: Use concepts of • Likely, unlikely, certain, certainty, fairness, and chance to impossible, improbable, discuss the probability of actual fair, unfair events. • More likely, less likely, equally likely 99 • Probability of tossing “heads” does not depend on outcomes of previous tosses 4.4.B.2 – Determine probabilities Pg. 384, No. 6: Determine the of simple events based on equally probability of a simple event likely outcomes and express them assuming equally likely as fractions. outcomes. 4.4.B.3 – Predict probabilities in Pg. 384, No. 7: Make predictions a variety of situations (e.g., given that are based on intuitive, the number of items of each color experimental, and theoretical in a bag, what is the probability probabilities. that an item picked will have a particular color). • What students think will happen (intuitive) • Collect data and use that data to predict the probability (experimental) • Analyze all possible outcomes to find the probability (theoretical) 4.4.C. Discrete Mathematics – Pgs. 452-453, Overview Systematic Listing and Counting 100 4.4.C.1 – Represent and classify Pg. 458, No. 4: Investigate ways data according to attributes, such to represent and classify data as shape or color, and according to attributes, such as relationships. shape or color, and relationships, • Venn diagrams and discuss the purpose and • Numerical and usefulness of such classification. alphabetical order 4.4.C.2 – Represent all Pg. 454, No. 1: Explore a variety possibilities for a simple counting of puzzles, games, and counting situation in an organized way and problems. draw conclusions from this representation. • Organized lists, charts, tree diagrams • Dividing into categories, (e.g., to find the total number of rectangles in a grid, find the number of rectangles of each size and add the results) 4.4.D. Discrete Mathematics – Vertex-Edge Graphs and Algorithms 4.4.D.1 – Follow, devise, and Pg. 458, No. 5: Follow, devise, describe practical sets of and describe practical lists of directions (e.g., to add two 2- instructions. digit numbers). 101 4.4.D.2 – Play two-person games Pg. 454, No. 1: Explore a variety and devise strategies for winning of puzzles, games, and counting the games (e.g., “make 5” where problems. players alternately add 1 or 2 and the person who reaches 5, or another designated number, is the winner). 4.4.D.3 – Explore vertex-edge graphs • Vertex, edge, neighboring/adjacent, number of neighbors • Path, circuit (i.e., path that ends at its starting point) 4.4.D.4 – Find the smallest Pg. 456, No. 2: Use networks number of colors needed to color and tree diagrams to represent a map or a graph. everyday situations. 102 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 103 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 104 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 105 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 106 MATHEMATICS GRADE 5 STANDARD STUDENT OUTCOME SUGGESTED ACTIVITIES TEACHER’S NOTES AND NJ FRAMEWORKS 1996 SUPPLEMENTARY RESOURCES 4.1 All students will develop 4.1.A. Number Sense Pgs. 190-191, Overview number sense and will perform standard numerical 4.1.A.1 – Use real-life Pg. 192, No. 11: Extend their operations and estimations on experiences, physical materials, understanding of the number all types of numbers in a and technology to construct system by constructing meanings variety of ways. meanings for numbers (unless for integers, rational numbers, 4.1.A. Number Sense otherwise noted, all indicators percents, exponents, roots, 4.1.B. Numerical Operations for grade 5 pertain to these sets absolute values, and numbers 4.1.C. Estimation of numbers as well). represented in scientific notation. • Demonstrate an understanding of negative numbers in real life settings (i.e., minus Cº, below sea level, football yardage) • All fractions as part of a whole, as a subset of a set, and as a location on a number line, and as divisions of whole numbers • Ratios • All decimals 107 4.1.A.2 – Recognize the decimal Pg. 192, No. 10: Understand nature of United States currency money notations, count and and compute with money. compute money, and recognize the decimal nature of United States currency. 4.1.A.3 – Demonstrate a sense of Pg. 193, No. 12: Develop the relative magnitudes of number sense necessary for numbers. estimation. • Standard, expanded, and Pg. 193, No. 13: Expand the written form to one sense of magnitudes of different million number types to include integers, rational numbers, and roots. 4.1.A.4 – Use whole numbers, Pg. 195, No. 18: Investigate the fractions, and decimals to relationships among fractions, represent equivalent forms of the decimals, and percents, and use same number. all of them appropriately. Pg 324, No. 9: Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation. 4.1.A.5 – Develop and apply Pg. 195, No. 17: Develop and number theory concepts in apply number theory concepts problem-solving situations. such as primes, factors, and • Primes, factors, multiples multiples, in real-world and mathematical problem situations. 4.1.A.6 – Compare and order Pg. 194, No. 15: Develop and numbers. use order relations for integers and rational numbers. 108 4.1.B. Numerical Operations Pg. 268, Overview 4.1.B.1 – Recognize the Pg. 269, No. 8: Extend student appropriate use of each understanding and use of arithmetic operation in problem arithmetic operations to fractions, situations. decimals, integers, and rational numbers. 4.1.B.2 – Construct, use, and Pg. 269, No. 6: Select and use explain procedures for appropriate computational performing addition and methods from mental math, subtraction with fractions and estimation, paper-and-pencil, and decimals with: calculator methods, and check • Pencil-and-paper the reasonableness of results. • Mental math • Calculator 4.1.B.3 – Use an efficient and Pg. 269, No. 8: Extend their accurate pencil-and-paper understanding and use of procedure for division of a 3- arithmetic operations to fractions, digit number by a 2-digit number. decimals, integers, and rational numbers. 4.1.B.4 – Select pencil-and- Pg. 269, No. 6: Select and use paper, mental math, or a appropriate computational calculator as the appropriate methods from mental math, computational method in a given estimation, paper-and-pencil, and situation depending on the calculator methods, and check context and numbers. the reasonableness of results. 109 4.1.B.5 – Check the Pg. 269, No. 6: Select and use reasonableness of results of appropriate computational computations. methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results. 4.1.B.6 – Understand and use the various relationships among operations and properties of operations. 4.1.C. Estimation Pg. 321, Overview 4.1.C.1 – Use a variety of Pg. 323, No. 8: Develop, apply, estimation strategies for both and explain a variety of different number and computation. estimation strategies in problem situations involving quantities and measurement. 4.1.C.2 – Recognize when an Pg. 322, No. 5: Recognize when estimate is appropriate, and estimation is appropriate, and understand the usefulness of an understand the usefulness of an estimate as distinct from an exact estimate as distinct from an exact answer. answer. 4.1.C.3 – Determine the Pg. 322, No. 6: Determine the reasonableness of an answer by reasonableness of an answer by estimating the result of estimating the result of operations. operations. 110 4.1.C.4 – Determine whether a Pg. 324, No. 10: Determine given estimate is an overestimate whether a given estimate is an or an underestimate. overestimate or an underestimate. 4.2 All students will develop 4.2.A. Geometric Properties Pgs. 219-220, Overview spatial sense and the ability to use geometric properties, 4.2.A.1 – Understand and apply Pg. 229, No. 14: Understand the relationships, and concepts involving lines and properties of lines and planes, measurement to model, angles. including parallel and describe and analyze • Notation for line, ray, perpendicular lines and planes, phenomena. angle, line segment and intersecting lines and planes 4.2.A. Geometric Properties • Properties of parallel, and their angles of incidence. 4.2.B. Transforming Shapes perpendicular, and 4.2.C. Coordinate Geometry intersecting lines 4.2.D. Units of Measurement • Sum of the measures of 4.2.E. Measuring Geometric the interior angles of a Objects triangle is 180º 4.2.A.2 – Identify, describe, Pg. 228, No. 13: Identify, compare, classify polygons, and describe, compare, and classify solid figures. plane and solid geometric • Triangles by angles and figures. sides • Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi • Polygons by number of sides • Equilateral, equiangular, regular 111 • All points equidistant from a given point form a circle 4.2.A.3 – Identify similar figures. 4.2.A.4 – Understand and apply Pg. 228, No. 12: Understand and the concepts of congruence and apply the concepts of symmetry, symmetry (line and rotational). similarity, and congruence. 4.2.B. Transforming Shapes 4.2.B.1 – Use a translation, a Pg. 229, No. 15: Explore the reflection, or a rotation to map relationships among geometric one figure onto another transformations (translations, congruent figure. reflections, rotations, and dilations), tessellations (tilings), and congruence and similarity. 4.2.B.2 – Recognize, identify, Pg. 231, No. 19: Investigate, and describe geometric explore, and describe the relationships and properties as geometry in nature and real- they exist in nature, art, and other world applications, using models, real-world settings. manipulatives, and appropriate technology. 4.2.C. Coordinate Geometry 4.2.C.1 – Create geometric Pg. 423, No. 9: Understand and shapes with specified properties use the rectangular coordinate in the first quadrant on a system. coordinate grid. 112 4.2.D. Units of Measurement Pgs. 294-295, Overview Pgs. 496-497, Overview 4.2.D.1 – Select and use Pg. 298, No. 14: Understand and appropriate units to measure apply measurement in their own angles and area. lives and in other subject areas. Pg. 230, No. 16: Develop, understand, and apply a variety of strategies for determining perimeter, area, surface area, angle measure, and volume. 4.2.D.2 – Convert measurement Pg. 296, No. 8: Read and units within a system (e.g., 3 feet interpret various scales, including = _____ inches). those based on number lines and maps. 4.2.D.3 – Know approximate Pg. 298, No. 13: Convert equivalents between the standard measurement units from one and metric systems (e.g., one form to another, and carry out kilometer is approximately 6/10 calculations that involve various of a mile). units of measurement. 4.2.D.4 – Use measurements and Pg. 298, No. 13: Convert estimates to describe and measurement units from one compare phenomena. form to another, and carry out calculations that involve various units of measurement. 113 Pg. 296, No. 7: Use estimated and actual measurements to describe and compare phenomena. 4.2.E. Measuring Geometric Pgs. 225-226, Overview Objects 4.2.E.1 – Use a protractor to measure angles. 4.2.E.2 – Develop and apply Pg. 230, No. 16: Develop, strategies and formulas for understand, and apply a variety finding perimeter, area, and of strategies for determining volume perimeter, area, surface area, • Square/cube angle measure, and volume. • Rectangle/rectangular Pg. 297, No. 11: Develop prism formulas and procedures for solving problems related to measurement. 4.2.E.3 – Recognize that rectangles with the same perimeter do not necessarily have the same area and vice versa. 4.2.E.4 – Develop informal ways Pg. 299, No. 15: Understand and of approximating the measures of explain the impact of the change familiar objects (e.g., use a grid of an object’s linear dimensions to approximate the area of the on its perimeter, area, or volume. bottom of one’s foot). 114 4.3 All students will represent 4.3.A. Patterns Pgs. 350-351, Overview and analyze relationships Pgs. 418-419, Overview among variable quantities and solve problems involving 4.3.A.1 – Recognize, describe, Pg. 231, No. 18: Explore patterns, functions, and extend, and create patterns patterns produced by processes of algebraic concepts and involving whole numbers. geometric change, relating processes. • Descriptions using tables, iteration, approximation, and 4.3.A. Patterns verbal rules, simple fractals. 4.3.B. Functions and equations, and graphs. Pg. 352, No. 7: Represent and Relationships describe mathematical 4.3.C. Modeling relationships with tables, rules, 4.3.D. Procedures simple equations, and graphs. Pg. 356, No. 13: Develop, analyze, and explain arithmetic sequences. 4.3.B. Functions and Relationships 4.3.B.1 – Describe arithmetic Pg. 355, No. 11: Understand and operations as functions, including describe the general behavior of combining operations and functions. reversing them. 4.3.B.2 – Graph points satisfying a function from T-charts, from verbal rules, and from simple equations. 4.3.C. Modeling 4.3.C.1 – Use number sentences Pg. 355, No. 12: Use patterns, to model situations. relationships, and linear functions 115 • Using variables to to model situations in represent unknown mathematics and in other areas. quantities Pg. 354, No. 9: Use patterns, • Using concrete materials, relationships, and functions to tables, graphs, verbal model situations and to solve rules, algebraic problems, in mathematics and in expressions/equations other subject areas. Pg. 353, No. 8: Understand and describe the relationships among various representations of patterns and functions. 4.3.C.2 – Draw freehand sketches Pg. 420, No. 6: Represent of graphs that model real situations and number patterns phenomena and use such graphs with concrete materials, tables, to predict and interpret events. graphs, verbal rules, and standard • Changes over time algebraic notation. • Rates of change (e.g., Pg. 425, No. 13: Draw freehand when is plant growing sketches of, and interpret, graphs slowly/rapidly, when is which model real phenomena. temperature dropping Pg. 354, No. 10: Analyze most rapidly/slowly) functional relationships to explain how a change in one quantity results in a change in another. 4.3.D. Procedures 4.3.D.1 – Solve simple linear Pg. 424, No. 10: Solve simple equations with manipulatives and linear equations using concrete, informally. informal, and graphical methods, as well as appropriate paper-and- 116 pencil techniques. • Whole-number coefficients only, answers also whole numbers • Variables on one side of equation 4.4 All students will develop an 4.4.A. Data Analysis Pgs. 350-351, Overview understanding of the concepts and techniques of data 4.4.A.1 – Collect, generate, Pg. 388, No. 9: Generate, analysis, probability, and organize, and display data. collect, organize, and analyze discrete mathematics, and • Data generated from data and represent this data in will use them to model surveys tables, charts, and graphs. situations, solve problems, and analyze and draw 4.4.A.2 – Read, interpret, select, Pg. 389, No. 11: Make appropriate inferences from construct, analyze, generate inferences and formulate and data. questions about, and draw evaluate arguments based on data 4.4.A. Data Analysis inferences from displays of data. analysis and data displays. 4.4.B. Probability • Bar graph, line graph, Pg. 388, No. 10: Select and use 4.4.C. Discrete Mathematics – circle graph, table appropriate graphical Systematic Listing and Counting • Range, median, and mean representations and measures of 4.4.D. Discrete Mathematics – central tendency (mean, mode, Vertex-Edge Graphs and and median) for sets of data. Algorithms 4.4.A.3 – Respond to questions Pg. 389, No. 12: Use lines of about data and generate their own best fit to interpolate and predict questions and hypotheses. from data. Pg. 389, No. 11: Make inferences and formulate and evaluate arguments based on data analysis and data displays. 117 4.4.B. Probability Pgs. 386-387, Overview 4.4.B.1 – Determine probabilities Pg. 391, No. 16: Interpret of events. probabilities as ratios and • Event, probability of an percents. event • Probability of certain event is 1 and of impossible event is 0 4.4.B.2 – Determine probability Pg. 389, No. 13: Determine the using intuitive, experimental, and probability of a compound event. theoretical methods (e.g., using Pg. 391, No. 16: Interpret model of picking items of probabilities as ratios and different colors from a bag). percents. • Given numbers of various Pg. 390, No. 15: Use models of types of items in a bag, probability to predict events what is the probability based on actual data. that an item of one type will be picked • Given data obtained experimentally, what is the likely distribution of items in the bag 4.4.B.3 – Model situations Pg. 390, No. 14: Model involving probability using situations involving probability, simulations (with spinners, dice) such as genetics, using both and theoretical models. simulations and theoretical models. 118 4.4.C. Discrete Mathematics – Pgs. 462-463, Overview Systematic Listing and Counting 4.4.C.1 – Solve counting Pg. 464, No. 6: Use systematic problems and justify that all listing, counting, and reasoning possibilities have been in a variety of different contexts. enumerated without duplication • Organized lists, charts, tree diagrams, tables 4.4.C.2 – Explore the multiplication principle of counting in simple situations by representing all possibilities in an organized way (e.g., you can make 3 X 4 = 12 outfits using 3 shirts and 4 skirts). 4.4.D. Discrete Mathematics – Vertex-Edge Graphs and Algorithms 4.4.D.1 – Devise strategies for Pg. 465, No. 7: Recognize winning simple games (e.g., start common discrete mathematical with two piles of objects, each of models, explore their properties, two players in turn removes any and design them for specific number of objects from a single situations. pile, and the person to take the last group of objects wins) and express those strategies as sets of directions. 119 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 120 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 121 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 122 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 123 MATHEMATICS GRADE 6 STANDARD STUDENT OUTCOME SUGGESTED ACTIVITIES TEACHER’S NOTES AND NJ FRAMEWORKS 1996 SUPPLEMENTARY RESOURCES 4.1 All students will develop 4.1.A. Number Sense Pgs. 190-191, Overview number sense and will perform standard numerical 4.1.A.1 – Use real-life Pg. 192, No. 11: Extend their operations and estimations experiences, physical materials, understanding of the number on all types of numbers in a and technology to construct system by constructing meanings variety of ways. meanings for numbers (unless for integers, rational numbers, 4.1.A. Number Sense otherwise noted, all indicators percents, exponents, roots, 4.1.B. Numerical Operations for grade 6 pertain to these sets absolute values, and numbers 4.1.C. Estimation of numbers as well). represented in scientific notation. • All integers • All fractions as part of a whole, as subset of a set, as a location on a number line, and as divisions of whole numbers • All decimals 4.1.A.2 – Recognize the decimal Pg. 192, No. 10: Understand nature of United States currency money notations, count and and compute with money. compute money, and recognize the decimal nature of United States currency. 4.1.A.3 – Demonstrate a sense of Pg. 193, No. 12: Develop the relative magnitudes of number sense necessary for numbers. estimation. 124 Pg. 193, No. 13: Expand the sense of magnitudes of different number types to include integers, rational numbers, and roots. 4.1.A.4 – Explore the use of Pg. 194, 14: Understand and ratios and proportions in a variety apply ratios, proportions, and of situations. percents in a variety of situations. Pg. 271, No. 11: Develop, apply and explain methods for solving problems involving proportions and percents. 4.1.A.5 – Understand and use Pg. 192, No. 11: Extend their whole-number percents between understanding of the number 1 and 100 in a variety of system by constructing meanings situations. for integers, rational numbers, percents, exponents, roots, absolute values, and numbers represented in scientific notation. Pg. 194, 14: Understand and apply ratios, proportions, and percents in a variety of situations. Pg. 271, No. 11: Develop, apply and explain methods for solving problems involving proportions and percents. 4.1.A.6 – Use whole numbers, Pg. 195, No. 18: Investigate the fractions, and decimals to relationships among fractions, represent equivalent forms of the decimals, and percents, and use same number. all of them appropriately. Pg 324, No. 9: Use equivalent 125 representations of numbers such as fractions, decimals, and percents to facilitate estimation. 4.1.A.7 – Develop and apply Pg. 195, No. 17: Develop and number theory concepts in apply number theory concepts, problem solving situations. such as, primes, factors, and • Primes, factors, multiples multiples, in real-world and • Common multiples, mathematical problem situations. common factors • Least common multiple, greatest common factor 4.1.A.8 – Compare and order Pg. 194, No. 15: Develop and numbers. use order relations for integers and rational numbers. 4.1.B. Numerical Operations Pg. 268, Overview 4.1.B.1 – Recognize the Pg. 269, No. 8: Extend their appropriate use of each understanding and use of arithmetic operation in problem arithmetic operations to fractions, situations. decimals, integers, and rational numbers. 4.1.B.2 – Construct, use, and Pg. 269, No. 6: Select and use explain procedures for appropriate computational performing calculations with methods from mental math, fractions and decimals with: estimation, paper-and-pencil, and • Pencil-and-paper calculator methods, and check • Mental math the reasonableness of results. • Calculator 126 4.1.B.3 – Use an efficient and Pg. 269, No. 8: Extend their accurate pencil-and-paper understanding and use of procedure for division of a 3- arithmetic operations to fractions, digit number by a 2-digit number. decimals, integers, and rational numbers. 4.1.B.4 – Select pencil-and- Pg. 269, No. 6: Select and use paper, mental math, or a appropriate computational calculator as the appropriate methods from mental math, computational method in a given estimation, paper-and-pencil, and situation depending on the calculator methods, and check context and numbers. the reasonableness of results. 4.1.B.5 – Find squares and cubes Pg. 270, No. 9: Extend their of whole numbers. understanding of basic arithmetic operations on whole numbers to include powers and roots. 4.1.B.6 – Check the Pg. 269, No. 6: Select and use reasonableness of results of appropriate computational computations. methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results. 4.1.B.7 – Understand and use the various relationships among operations and properties of operations. 127 4.1.B.8 – Understand and apply Pg. 272, No. 12: Understand and the standard algebraic order of apply the standard algebraic operations for the four basic order of operations. operations, including appropriate use of parentheses. 4.1.C. Estimation Pg. 321, Overview 4.1.C.1 – Use a variety of Pg. 323, No. 8: Develop, apply, strategies for estimating both and explain a variety of different quantities and the results of estimation strategies in problem computations. situations involving quantities and measurement. 4.1.C.2 – Recognize when an Pg. 322, No. 5: Recognize when estimate is appropriate, and estimation is appropriate, and understand the usefulness of an understand the usefulness of an estimate as distinct from an exact estimate as distinct from an exact answer. answer. 4.1.C.3 – Determine the Pg. 322, No. 6: Determine the reasonableness of an answer by reasonableness of an answer by estimating the result of estimating the result of operations. operations. 4.1.C.4 – Determine whether a Pg. 324, No. 10: Determine given estimate is an overestimate whether a given estimate is an or an underestimate. overestimate or an underestimate. 128 4.2 All students will develop 4.2.A. Geometric Properties Pgs. 219-220, Overview spatial sense and the ability to use geometric properties, 4.2.A.1 – Understand and apply Pg. 229, No. 14: Understand the relationships, and concepts involving lines and properties of lines and planes, measurement to model, angles. including parallel and describe and analyze • Notation for line, ray, perpendicular lines and planes, phenomena. angle, line segment and intersecting lines and planes 4.2.A. Geometric Properties • Properties of parallel, and their angles of incidence. 4.2.B. Transforming Shapes perpendicular, and 4.2.C. Coordinate Geometry intersecting lines 4.2.D. Units of Measurement • Sum of the measurements 4.2.E. Measuring Geometric of the interior angles of a Objects triangle is 180˚ 4.2.A.2 – Identify, describe, Pg. 228, No. 13: Identify, compare, and classify polygons describe, compare, and classify and circles. plane and solid geometric • Triangles by angles and figures. sides • Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi • Polygons by number of sides • Equilateral, equiangular, regular • All points equidistant from a given point form a circle 129 4.2.A.3 – Identify similar figures. 4.2.A.4 – Understand and apply Pg. 228, No. 12: Understand and the concepts of congruence and apply the concepts of symmetry, symmetry (line and rotational). similarity, and congruency. 4.2.A.5 – Compare properties of Pg. 228, No. 13: Identify, cylinders, prisms, cones, describe, compare, and classify pyramids, and spheres. plane and solid geometric figures. 4.2.A.6 – Identify, describe, and Pg. 227, No. 11: Relate two- draw the faces or shadows dimensional and three- (projections) of three- dimensional geometry using dimensional geometric objects shadows, perspectives, from different perspectives. projections, and maps. 4.2.A.7 – Identify a three- Pg. 227, No. 11: Relate two- dimensional shape with given dimensional and three- projections (top, front and side dimensional geometry using views). shadows, perspectives, projections, and maps. 4.2.A.8 – Identify a three- dimensional shape with a given net (i.e., a flat pattern that folds into a 3D shape). 130 4.2.B. Transforming Shapes 4.2.B.1 – Use a translation, a Pg. 229, No. 15: Explore the reflection, or a rotation to map relationships among geometric one figure onto another transformations (translations, congruent figure. reflections, rotations, and dilations), tessellations (tilings), and congruency and similarity. 4.2.B.2 – Recognize, identify, Pg. 231, No. 19: Investigate, and describe geometric explore, and describe the relationships and properties as geometry in nature and real- they exist in nature, art, and other world applications, using models, real-world settings. manipulatives, and appropriate technology. 4.2.C. Coordinate Geometry 4.2.C.1 – Create geometric Pg. 423, No. 9: Understand and shapes with specified properties use the rectangular coordinate in the first quadrant on a system. coordinate grid. 4.2.D. Units of Measurement Pgs. 294-295, Overview Pgs. 496-497, Overview 4.2.D.1 – Select and use Pg. 298, No. 14: Understand and appropriate units to measure apply measurement in their own angles, area, surface area, and lives and in other subject areas. volume. Pg. 230, No. 16: Develop, understand, and apply a variety of strategies for determining 131 perimeter, area, surface area, angle measure, and volume. 4.2.D.2 – Use a scale to find a Pg. 296, No. 8: Read and distance on a map or a length on interpret various scales, including a scale drawing. those based on number lines and maps. 4.2.D.3 – Convert measurement Pg. 298, No. 13: Convert units within a system (e.g., 3 feet measurement units from one = ____ inches). form to another, and carry out calculations that involve various units of measurement. 4.2.D.4 – Know approximate Pg. 298, No. 13: Convert equivalents between the standard measurement units from one and metric systems (e.g., one form to another, and carry out kilometer is approximately 6/10 calculations that involve various of a mile). units of measurement. 4.2.D.5 – Use measurements and Pg. 296, No. 7: Use estimated estimates to describe and and actual measurements to compare phenomena. describe and compare phenomena. 4.2.E. Measuring Geometric Pgs. 225-226, Overview Objects 4.2.E.1 – Use a protractor to measure angles. 132 4.2.E.2 – Develop and apply Pg. 230, No. 16: Develop, strategies and formulas for understand, and apply a variety finding perimeter and area. of strategies for determining • Triangle, square, perimeter, area, surface area, rectangle, parallelogram, angle measure, and volume. and trapezoid Pg. 297, No. 11: Develop • Circumference and area formulas and procedures for of a circle solving problems related to measurement. 4.2.E.3 – Develop and apply Pg. 230, No. 16: Develop, strategies and formulas for understand, and apply a variety finding the surface area and of strategies for determining volume of rectangular prisms and perimeter, area, surface area, cylinders. angle measure, and volume. Pg. 297, No. 11: Develop formulas and procedures for solving problems related to measurement. 4.2.E.4 – Recognize that shapes Pg. 299, No. 15: Understand and with the same perimeter do not explain the impact of the change necessarily have the same area of an object’s linear dimensions and vice versa. on its perimeter, area, or volume. 4.2.E.5 – Develop informal ways Pg. 297, No. 12: Explore of approximating the measures of situations involving quantities familiar objects (e.g., use a grid which cannot be measured to approximate the area of the directly or conveniently. bottom of one’s foot). 133 4.3 All students will represent 4.3.A. Patterns Pgs. 350-351, Overview and analyze relationships Pgs. 418-419, Overview among variable quantities and solve problems 4.3.A.1 – Recognize, describe, Pg. 195, No. 16: Recognize and involving patterns, functions, extend, and create patterns describe patterns in both finite and algebraic concepts and involving whole numbers and and infinite number sequences processes. rational numbers. involving whole numbers, 4.3.A. Patterns • Descriptions using tables, rational numbers, and integers. 4.3.B. Functions and verbal rules, simple Pg. 231, No. 18: Explore Relationships equations, and graphs patterns produced by processes of 4.3.C. Modeling • Formal iterative formulas geometric change, relating 4.3.D. Procedures (e.g., NEXT = NOW * 3) iteration, approximation, and • Recursive patterns, fractals. including Pascal’s Pg. 352, No. 7: Represent and Triangle (where each describe mathematical entry is the sum of the relationships with tables, rules, entries above it) and the simple equations, and graphs. Fibonacci Sequence: 1, 1, Pg. 356, No. 13: Develop, 2, 3, 5, 8, … (where analyze, and explain arithmetic NEXT = NOW + sequences. PREVIOUS) Pg. 466, No. 8: Experiment with iterative and recursive processes, with the aid of calculators and computers. 4.3.B. Functions and Relationships 4.3.B.1 – Describe the general Pg. 355, No. 11: Understand and behavior of functions given by describe the general behavior of formulas or verbal rules (e.g., functions. graph to determine whether increasing or decreasing, linear or not). 134 4.3.C. Modeling 4.3.C.1 – Use patterns, relations, Pg. 353, No. 8: Understand and and linear functions to model describe the relationships among situations. various representations of • Using variables to patterns and functions. represent unknown Pg. 354, No. 9: Use patterns, quantities relationships, and functions to • Using concrete materials, model situations and to solve tables, graphs, verbal problems, in mathematics and in rules, algebraic other subject areas. expressions/equations/in- Pg. 355, No. 12: Use patterns, equalities relationships, and linear functions to model situations in mathematics and in other areas. 4.3.C.2 – Draw freehand sketches Pg. 354, No. 10: Analyze of graphs that model real functional relationships to phenomena and use such graphs explain how a change in one to predict and interpret events. quantity results in a change in • Changes over time another. • Relations between Pg. 420, No. 6: Represent quantities situations and number patterns • Rates of change (e.g., with concrete materials, tables, when is plant growing graphs, verbal rules, and standard slowly/rapidly, when is algebraic notation. temperature dropping Pg. 425, No. 13: Draw freehand most rapidly/slowly) sketches of, and interpret, graphs which model real phenomena. 135 4.3.D. Procedures 4.3.D.1 – Solve simple linear Pg. 424, No. 10: Solve simple equations with manipulatives and linear equations using concrete, informally. informal, and graphical methods, • Whole-number as well as appropriate paper-and- coefficients only, answers pencil techniques. also whole numbers • Variables on one or both sides of equation 4.3.D.2 – Understand and apply the properties of operations and numbers. • Distributive property • The product of a number and its reciprocal is 1 4.3.D.3 – Evaluate numerical Pg. 420, No. 5: Understand and expressions. use variables, expressions, equations, and inequalities. 4.3.D.4 – Extend understanding Pg. 420, No. 5: Understand and and use of inequality. use variables, expressions, • Symbols (≥, ≠, ≤) equations, and inequalities. Pg. 424, No. 12: Investigate inequalities and nonlinear equations informally. 4.4 All students will develop an 4.4.A. Data Analysis understanding of the concepts and techniques of 4.4.A.1 – Collect, generate, Pg. 388, No. 9: Generate, data analysis, probability, organize, and display data. collect, organize, and analyze 136 and discrete mathematics, • Data generated from data and represent this data in and will use them to model surveys tables, charts, and graphs. situations, solve problems, and analyze and draw 4.4.A.2 – Read, interpret, select, Pg. 388, No. 10: Select and use appropriate inferences from construct, analyze, generate appropriate graphical data. questions about, and draw representations and measures of 4.4.A. Data Analysis inferences from displays of data. central tendency (mean, mode, 4.4.B. Probability • Bar graph, line graph, and median) for sets of data. 4.4.C. Discrete Mathematics – circle graph, table, Pg. 389, No. 11: Make Systematic Listing and Counting histogram inferences and formulate and 4.4.D. Discrete Mathematics – • Range, median, and mean evaluate arguments based on data Vertex-Edge Graphs and • Calculators and analysis and data displays. Algorithms computers used to record and process information 4.4.A.3 – Respond to questions Pg. 389, No. 11: Make about data, generate their own inferences and formulate and questions and hypotheses, and evaluate arguments based on data formulate strategies for analysis and data displays. answering their questions and Pg. 389, No. 12: Use lines of testing their hypotheses. best fit to interpolate and predict from data. 4.4.B. Probability Pgs. 386-387, Overview 4.4.B.1 – Determine probabilities Pg. 391, No. 16: Interpret of events. probabilities as ratios and • Event, complementary percents. event, probability of an event • Multiplication rule for probabilities • Probability of certain 137 event is 1 and of impossible event is 0 • Probability of event and complementary event add up to 1 4.4.B.2 – Determine probability Pg. 390, No. 15: Use models of using intuitive, experimental, and probability to predict events theoretical methods (e.g., using based on actual data. model of picking items of Pg. 391, No. 16: Interpret different colors from a bag). probabilities as ratios and • Given numbers of various percents. types of items in a bag, what is the probability that an item of one type will be picked • Given data obtained experimentally, what is the likely distribution of items in the bag 4.4.B.3 – Explore compound Pg. 389, No. 13: Determine the events. probability of a compound event. 4.4.B.4 – Model situations Pg. 390, No. 14: Model involving probability using situations involving probability, simulations (with spinners, dice) such as genetics, using both and theoretical models. simulations and theoretical models. 4.4.B.5 – Recognize and understand the connections among the concepts of 138 independent outcomes, picking at random, and fairness. 4.4.C. Discrete Mathematics – Pgs. 462-463, Overview Systematic Listing and Counting 4.4.C.1 – Solve counting Pg. 464, No. 6: Use systematic problems and justify that all listing, counting, and reasoning possibilities have been in a variety of different contexts. enumerated without duplication. • Organized lists, charts, three diagrams, tables • Venn diagrams 4.4.C.2 – Apply the multiplication principle of counting. • Simple situations (e.g., you can make 3 X 4 = 12 outfits using 3 shirts and 4 skirts) • Number of ways a specified number of items can be arranged in order (concept of permutation) • Number of ways of selecting a slate of officers from a class (e.g., if there are 23 students and 3 officers, the number is 23 X 22 X 21) 139 4.4.C.3 – List the possible combinations of two elements chosen from a given set (e.g., forming a committee of two from a group of 12 students, finding how many handshakes there will be among ten people if everyone shakes each other person’s hand once). 4.4.D. Discrete Mathematics – Vertex-Edge Graphs and Algorithms 4.4.D.1 – Devise strategies for Pg. 465, No. 7: Recognize winning simple games (e.g., start common discrete mathematical with two piles of objects, each of models, explore their properties, two players in turn removes any and design them for specific number of objects from a single situations. pile, and the person to take the last group of objects wins) and express those strategies as sets of directions. 4.4.D.2 – Analyze vertex-edge graphs and tree diagrams. • Can a picture or a vertex- edge graph be drawn with a single line? (degree of vertex) • Can you get from any vertex to any other vertex? (connectedness) 140 4.4.D.3 – Use vertex-edge graphs to find solutions to practical problems. • Delivery route that stops at specified sites but involves least travel • Shortest route from one site on a map to another 141 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 142 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 143 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 144 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 145 MATHEMATICS GRADE 7 STANDARD STUDENT OUTCOME SUGGESTED ACTIVITIES TEACHER’S NOTES AND NJ FRAMEWORKS 1996 SUPPLEMENTARY RESOURCES 4.1 All students will develop 4.1.A. Number Sense Pgs. 197-198, Overview number sense and will perform standard numerical 4.1.A.1 – Extend understanding Pg. 199, No. 11: Extend their operations and estimations on of the number system by understanding of the number all types of numbers in a constructing meanings for the system by constructing meanings variety of ways. following (unless otherwise for integers, rational numbers, 4.1.A. Number Sense noted, all indicators for grade 7 percents, exponents, roots, 4.1.B. Numerical Operations pertain to these sets of numbers absolute values, and numbers 4.1.C. Estimation as well). represented in scientific notation. • Rational numbers Pg. 201, No. 15: Develop and • Percents use order relations for integers • Whole numbers with and rational numbers. exponents 4.1.A.2 – Demonstrate a sense of Pg. 200, No. 13: Expand the the relative magnitudes of sense of magnitudes of different numbers. number types to include integers, rational numbers, and roots. 4.1.A.3 – Understand and use Pg. 201, No. 14: Understand and ratios, proportions, and percents apply ratios, proportions, and (including percents greater than percents in a variety of situations. 100 and less than 1) in a variety Pg. 275, No. 11: Develop, apply of situations. and explain the methods for solving problems involving proportions and percents. 146 Pg. 303, No. 12: Explore situations involving quantities which cannot be measured directly or conveniently. 4.1.A.4 – Compare and order Pg. 203, No. 19: Identify, derive, numbers of all named types. and compare properties of numbers. 4.1.A.5 – Use whole numbers, Pg. 199, No. 10: Understand fractions, decimals, and percents money notations, count and to represent equivalent forms of compute money, and recognize the same number. the decimal nature of United States currency. Pg. 203, No. 18: Investigate the relationships among fractions, decimals, and percents, and use all of them appropriately. Pg. 274, No. 8: Extend their understanding and use of arithmetic operations to fractions, decimals, integers, and rational numbers. Pg. 328, No. 9: Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation. 4.1.A.6 – Understand that all Pg. 202, No. 17: Develop and fractions can be represented as apply number theory concepts, repeating or terminating such as primes, factors, and decimals. multiples, in real-world and 147 mathematical problem situations. Pg. 505, No. 6: Investigate, represent, and use non- terminating decimals. 4.1.B. Numerical Operations Pg. 273, Overview 4.1.B.1 – Use and explain Pg. 274, No. 6: Select and use procedures for performing appropriate computational calculations with integers and all methods from mental math, number types named above with: estimation, paper-and-pencil, and • Pencil-and-paper calculator methods, and check • Mental math the reasonableness of results. • Calculator 4.1.B.2 – Use exponentiation to Pg. 274, No. 9: Extend their find whole number powers of understanding of basic arithmetic numbers. operations on whole numbers to include powers and roots. 4.1.B.3 – Understand and apply the standard algebraic order of operations, including appropriate use of parentheses. 4.1.C. Estimation Pg. 325, Overview 4.1.C.1 – Use equivalent Pg. 200, No. 12: Develop representations of numbers such number sense necessary for as fractions, decimals, and estimation. percents to facilitate estimation. 148 Pg. 275, No. 10: Develop, apply, and explain procedures for computation and estimation with whole numbers, fractions, decimals, integers, and rational numbers. 4.2 All students will develop 4.2.A. Geometric Properties Pgs. 233-234, Overview spatial sense and the ability to use geometric properties, 4.2.A.1 – Understand and apply relationships, and properties of polygons. measurement to model, • Quadrilaterals, including describe and analyze squares, rectangles, phenomena. parallelograms, 4.2.A. Geometric Properties trapezoids, rhombi 4.2.B. Transforming Shapes • Regular polygons 4.2.C. Coordinate Geometry 4.2.D. Units of Measurement 4.2.A.2 – Understand and apply Pg. 235, No. 12: Understand and 4.2.E. Measuring Geometric the concept of similarity. apply the concepts of symmetry, Objects • Using proportions to find similarity, and congruence. missing measures Pg. 302, No. 8: Read and • Scale drawings interpret various scales, including • Models of 3D objects those based on number lines and maps. Pg. 235, No. 11: Relate two- dimensional and three- dimensional geometry using shadows, perspectives, projections, and maps. Pg. 236, No. 13: Identify, describe, compare, and classify plane and solid geometric figures. 149 Pg. 305, No. 16: Apply their knowledge of measurement to the construction of a variety of two- and three-dimensional figures. 4.2.A.3 – Use logic and reasoning to make and support conjectures about geometric objects. 4.2.B. Transforming Shapes 4.2.B.1 – Understand and apply Pg. 237, No. 15: Explore the transformations. relationships among geometric • Finding the image, given transformations (translations, the pre-image, and vice reflections, rotations, and versa dilations), tessellations (tilings), • Sequence of and congruence and similarity. transformations needed to map one figure onto another • Reflections, rotations, and translations result in images congruent to the pre-image • Dilations (stretching/shrinking) result in images similar to the pre-image 150 4.2.C. Coordinate Geometry 4.2.C.1 – Use coordinates in four Pg. 431, No. 9: Understand and quadrants to represent geometric use the rectangular coordinate concepts. system. 4.2.C.2 – Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units). 4.2.D. Units of Measurement Pg. 301, Overview 4.2.D.1 – Solve problems requiring calculations that involve different units of measurement within a measurement system (e.g., 4’3” plus 7’10” equals 12”1”). 4.2.D.2 – Select and use Pg. 302, No. 9: Determine the appropriate units and tools to degree of accuracy needed in a measure quantities to the degree given situation and choose units of precision needed in a accordingly. particular problem-solving situation. 4.2.D.3 – Recognize that all Pg. 302, No. 10: Understand that measurements of continuous all measurements of continuous quantities are approximations. quantities are approximate. Pg. 506, No. 8: Approximate quantities with increasing degrees of accuracy. 151 4.2.E. Measuring Geometric Objects 4.2.E.1 – Develop and apply Pg. 238, No. 16: Develop, strategies for finding perimeter understand, and apply a variety and area. of strategies for determining • Geometric figures made perimeter, area, surface area, by combining triangles, angle measure, and volume. rectangles and circles or Pg. 302, No. 7: Use estimated parts of circles and actual measurements to • Estimation of area using describe and compare grids of various sizes phenomena. 4.2.E 2 – Recognize that the volume of a pyramid or cone is one-third of the volume of the prism or cylinder with the same base and height (e.g., use rice to compare volumes of figures with the same base and height. 4.3 All students will represent 4.3.A. Patterns Pg. 357, Overview and analyze relationships among variable quantities and 4.3.A.1 – Recognize, describe, Pg. 358, No. 8: Understand and solve problems involving extend, and create patterns describe the relationships among patterns, functions, and involving whole numbers, various representations of algebraic concepts and rational numbers, and integers. patterns and functions. processes. • Descriptions using tables, Pg. 202, No. 16: Recognize and 4.3.A. Patterns verbal and symbolic rules, describe patterns in both finite 4.3.B. Functions and graphs, simple equations and infinite number sequences Relationships or expressions involving whole numbers, 4.3.C. Modeling • Finite and infinite rational numbers, and integers. 4.3.D. Procedures sequences 152 • Generating sequences by Pg. 504, No. 5: Develop an using calculators to understanding of infinite repeatedly apply a sequences that arise in natural formula situations. Pg. 362, No. 13: Develop, analyze, and explain arithmetic sequences. 4.3.B. Functions and Pgs. 427-428, Overview Relationships 4.3.B.1 – Graph functions, and Pg. 360, No. 11: Understand and understand and describe their describe the general behavior of general behavior. functions. • Equations involving two Pg. 430, No. 8: Analyze tables variables and graphs to identify properties and relationships. 4.3.C. Modeling 4.3.C.1 – Analyze functional Pg. 360, No. 10: Analyze relationships to explain how a functional relationships to change in one quantity can result explain how a change in one in a change in another, using quantity results in a change in pictures, graphs, charts, and another. equations. Pg. 505, No. 7: Represent, analyze, and predict relations between quantities, especially quantities changing over time. 153 4.3.C.2 – Use patterns, relations, Pg. 304, No. 15: Understand and symbolic algebra, and linear explain the impact of the change functions to model situations. of an object’s linear dimensions • Using manipulatives, on its perimeter, area, or volume. tables, graphs, verbal Pg. 358, No. 7: Represent and rules, algebraic describe mathematical expressions/equations/in- relationships with tables, rules, equalities simple equations, and graphs. • Growth situations, such as Pg. 361, No. 12: Use patterns, population growth and relationships, and linear functions compound interest, using to model situations in recursive (e.g., NOW- mathematics and in other areas. NEXT) formulas (cf. Pg. 429, No. 5: Understand and science standard 5.5 and use variables, expressions, social studies standard equations, and inequalities. 6.6) Pg. 429, No. 6: Represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and standard algebraic notation. Pg. 504, No. 4: Recognize and express the difference between linear and exponential growth. 4.3.D. Procedures 4.3.D.1 – Use graphing Pg. 430, No. 7: Use graphing techniques on a number line. techniques on a number line to • Absolute value model both absolute value and • Arithmetic operations arithmetic operations. represented by vectors Pg. 432, No. 12: Investigate (arrows) (e.g., “-3 + 6” is inequalities and nonlinear “left 3, right 6”) equations informally. 154 4.3.D.2 – Solve simple linear Pg. 432, No. 10: Solve simple equations informally and linear equations using concrete, graphically. informal, and graphical methods, • Multi-step, integer as well as appropriate paper-and- coefficients only pencil techniques. (although answers may Pg. 432, No. 11: Explore linear not be integers) equations through the use of • Using paper-and-pencil, calculators, computers, and other calculators, graphing technology. calculators, spreadsheets, and other technology 4.3.D.3 – Create, evaluate, and simplify algebraic expressions involving variables. • Order of operations, including appropriate use of parentheses • Substitution of a number for a variable 4.3.D.4 – Understand and apply Pg. 203, No. 19: Identify, derive, the properties of operations, and compare properties of numbers, equations, and numbers. inequalities Pg. 432, No. 12: Investigate • Additive inverse inequalities and nonlinear • Multiplicative inverse equations informally. 4.4 All students will develop an 4.4.A. Data Analysis understanding of the concepts and techniques of data 4.4.A.1 – Select and use Pg. 394, No. 9: Generate, analysis, probability, and appropriate representations for collect, organize, and analyze 155 discrete mathematics, and sets of data, and measures of data and represent this data in will use them to model central tendency (mean, median, tables, charts, and graphs. situations, solve problems, and mode). Pg. 394, No. 10: Select and use and analyze and draw • Type of display most appropriate graphical appropriate inferences from appropriate for given data representations and measures of data. • Box-and-whisker plot, central tendency (mean, mode, 4.4.A. Data Analysis upper quartile, lower and median) for sets of data. 4.4.B. Probability quartile 4.4.C. Discrete Mathematics – • Scatter plot Systematic Listing and Counting • Calculators and computer 4.4.D. Discrete Mathematics – used to record and Vertex-Edge Graphs and process information Algorithms 4.4.A.2 – Make inferences and Pg. 395, No. 11: Make formulate and evaluate inferences and formulate and arguments based on displays and evaluate arguments based on data analysis of data. analysis and data displays. 4.4.B. Probability Pgs. 392-393, Overview 4.4.B.1 – Interpret probabilities Pg. 396, No. 16: Interpret as ratios, percents, and decimals. probabilities as ratios and percents. 4.4.B.2 – Model situations Pg. 396, No. 15: Use models of involving probability with probability to predict events simulations (using spinners, dice, based on actual data. calculators and computers) and theoretical models. • Frequency, relative frequency 156 4.4.B.3 – Estimate probabilities Pg. 396, No. 14: Model and make predictions based on situations involving probability, experimental and theoretical such as genetics, using both probabilities. simulations and theoretical methods. 4.4.B.4 – Play and analyze probability-based games, and discuss the concepts of fairness and expected value. 4.4.C. Discrete Mathematics – Pgs. 471-472, Overview Systematic Listing and Counting 4.4.C.1 – Apply the Pg. 473, No. 6: Use systematic multiplication principle of listing, counting, and reasoning counting. in a variety of different contexts. • Permutations: ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23-student class) 4.4.C.2 – Explore counting Pg. 476, No. 9: Explore methods problems involving Venn for storing, processing, and diagrams with three attributes communicating information. (e.g., there are 15, 20, and 25 students respectively in the chess 157 club, the debating team, and the engineering society; how many different students belong to the three clubs if there are 6 students in chess and debating, 7 students in chess and engineering, 8 students in debating and engineering, and 2 students in all three?). 4.4.C.3 – Apply techniques of Pg. 359, No. 9: Use patterns, systematic listing, counting, and relationships, and functions to reasoning in a variety of different model situations and to solve contexts. problems in mathematics and in other subject areas. 4.4.D. Discrete Mathematics – Vertex-Edge Graphs and Algorithms 4.4.D.1 – Use vertex-edge graphs Pg. 474, No. 7: Recognize to represent and find solutions to common discrete mathematical practical problems. models, explore their properties, • Finding the shortest and design them for specific network connecting situations. specified sites Pg. 477, No. 10: Devise, • Finding the shortest route describe, and test algorithms for on a map from one site to solving optimization and search another problems. • Finding the shortest circuit on a map that makes a tour of specified sites 158 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 159 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 160 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 161 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 162 MATHEMATICS GRADE 8 STANDARD STUDENT OUTCOME SUGGESTED ACTIVITIES TEACHER’S NOTES AND NJ FRAMEWORKS 1996 SUPPLEMENTARY RESOURCES 4.1 All students will develop 4.1.A. Number Sense Pgs. 197-198, Overview number sense and will perform standard numerical 4.1.A.1 – Extend understanding Pg. 199, No. 11: Extend their operations and estimations on of the number system by understanding of the number all types of numbers in a constructing meanings for the system by constructing meanings variety of ways. following (unless otherwise for integers, rational numbers, 4.1.A. Number Sense noted, all indicators for grade 8 percents, exponents, roots, 4.1.B. Numerical Operations pertain to these sets of numbers absolute values, and numbers 4.1.C. Estimation as well). represented in scientific notation. • Rational numbers • Percents • Exponents • Roots • Absolute values • Numbers represented in scientific notation 4.1.A.2 – Demonstrate a sense of Pg. 200, No. 13: Expand the the relative magnitudes of sense of magnitudes of different numbers. number types to include integers, rational numbers, and roots. 4.1.A.3 – Understand and use Pg. 201, No. 14: Understand and ratios, rates, proportions, and apply ratios, proportions, and percents (including percents percents in a variety of situations. 163 greater than 100 and less than 1) Pg. 275, No. 11: Develop, apply in a variety of situations. and explain the methods for solving problems involving proportions and percents. Pg. 303, No. 12: Explore situations involving quantities which cannot be measured directly or conveniently. 4.1.A.4 – Compare and order Pg. 203, No. 19: Identify, derive, numbers of all named types. and compare properties of numbers. 4.1.A.5 – Use whole numbers, Pg. 199, No. 10: Understand fractions, decimals, and percents money notations, count and to represent equivalent forms of compute money, and recognize the same number. the decimal nature of United States currency. Pg. 203, No. 18: Investigate the relationships among fractions, decimals, and percents, and use all of them appropriately. Pg. 274, No. 8: Extend their understanding and use of arithmetic operations to fractions, decimals, integers, and rational numbers. Pg. 328, No. 9: Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation. 164 4.1.A.6 – Recognize that repeating decimals correspond to fractions and determine their fractional equivalents. • 5/7 = 0.714285714285… = 0. Pg. 505, No. 6: Investigate, 4.1.A.7 – Construct meanings for represent, and use non- common irrational numbers, such terminating decimals. as π (pi) and the square root of 2. Pg. 273, Overview 4.1.B. Numerical Operations Pg. 274, No. 6: Select and use 4.1.B.1 – Use and explain appropriate computational procedures for performing methods from mental math, calculations involving addition, estimation, paper-and-pencil, and subtraction, multiplication, calculator methods, and check division, and exponentiation with the reasonableness of results. integers and all number types named above with: • Pencil-and-paper • Mental math • Calculator Pg. 274, No. 9: Extend their 4.1.B.2 – Use exponentiation to understanding of basic arithmetic find whole number powers of operations on whole numbers to numbers. include powers and roots. 4.1.B.3 – Find square and cube roots of numbers and understand the inverse nature of powers and roots. 165 4.1.B.4 – Solve problems Pg. 275, No. 11: Develop, apply, involving proportions and and explain methods for solving percents. problems involving proportions and percents. 4.1.B.5 – Understand and apply Pg. 276, No. 12: Understand and the standard algebraic order of apply the standard algebraic operations, including appropriate order of operations. use of parentheses. 4.1.C. Estimation Pg. 325, Overview 4.1.C.1 – Estimate square and cube roots of numbers. 4.1.C.2 – Use equivalent Pg. 200, No. 12: Develop representations of numbers such number sense necessary for as fractions, decimals, and estimation. percents to facilitate estimation. Pg. 275, No. 10: Develop, apply, and explain procedures for computation and estimation with whole numbers, fractions, decimals, integers, and rational numbers. 4.1.C.3 – Recognize the Pg. 326, No. 5: Recognize when limitations of estimation and estimation is appropriate, and assess the amount of error understand the usefulness of an resulting from estimation. estimate as distinct from an exact answer. 166 Pg. 326, No. 6: Determine the reasonableness of an answer by estimating the result of operations. Pg. 328, No. 10: Determine whether a given estimate is an overestimate or an underestimate. 4.2 All students will develop 4.2.A. Geometric Properties Pgs. 233-234, Overview spatial sense and the ability to use geometric properties, 4.2.A.1 – Understand and apply Pg. 237, No. 14: Understand the relationships, and concepts involving lines, angles, properties of lines and planes, measurement to model, and planes. including parallel and describe and analyze • Complementary and perpendicular lines and planes, phenomena. supplementary angles and intersecting lines and planes 4.2.A. Geometric Properties • Vertical angles and their angles of incidence. 4.2.B. Transforming Shapes • Bisectors and 4.2.C. Coordinate Geometry perpendicular bisectors 4.2.D. Units of Measurement • Parallel, perpendicular, 4.2.E. Measuring Geometric and intersecting planes Objects • Intersection of plane with cube, cylinder, cone, and sphere 4.2.A.2 – Understand and apply Pg. 238, No. 17: Understand and the Pythagorean Theorem. apply the Pythagorean Theorem. 4.2.A.3 – Understand and apply properties of polygons. • Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi 167 • Regular polygons • Sum of measures of interior angles of a polygon • Which polygons can be used alone to generate a tessellation and why 4.2.A.4 – Understand and apply Pg. 303, No. 12: Explore the concept of similarity. situations involving quantities • Using proportions to find which cannot be measured missing measures directly or conveniently. • Scale drawings Pg. 302, No. 8: Read and • Models of 3D objects interpret various scales, including those based on number lines and maps. Pg. Pg. 235, No. 11: Relate two- dimensional and three- dimensional geometry using shadows, perspectives, projections, and maps. Pg. 236, No. 13: Identify, describe, compare, and classify plane and solid geometric figures. Pg. 305, No. 16: Apply their knowledge of measurement to the construction of a variety of two- and three-dimensional figures. 168 4.2.A.5 – Use logic and reasoning to make and support conjectures about geometric objects. 4.2.A.6 – Perform basic geometric constructions using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology). • Congruent angles or line segments • Midpoint of a line segment 4.2.A.7 – Create two- dimensional representations (e.g., nets or projective views) for the surfaces of three-dimensional objects. 4.2.B. Transforming Shapes 4.2.B.1 – Understand and apply Pg. 237, No. 15: Explore the transformations. relationships among geometric • Finding the image, given transformations (translations, the pre-image, and vice reflections, rotations, and versa dilations), tessellations (tilings), • Sequence of and congruence and similarity. transformations needed to map one figure onto 169 another • Reflections, rotations, and translations result in images congruent to the pre-image • Dilations (stretching/shrinking) result in images similar to the pre-image 4.2.B.2 – Use iterative Pg. 238, No. 18: Explore procedures to generate geometric patterns produced by processes of patterns. geometric change, relating • Fractals (e.g., the Koch iteration, approximation, and Snowflake) fractals. • Self-similarity • Construction of initial stages • Patterns in successive stages (e.g., number of triangles in each stage of Sierpinski’s Triangle) 4.2.C. Coordinate Geometry 4.2.C.1 – Use coordinates in four Pg. 431, No. 9: Understand and quadrants to represent geometric use the rectangular coordinate concepts. system. 4.2.C.2 – Use a coordinate grid to model and quantify 170 transformations (e.g., translate right 4 units). 4.2.D. Units of Measurement Pg. 301, Overview 4.2.D.1 – Solve problems Pg. 303, No. 13: Convert requiring calculations that measurement units from one involve different units of form to another, and carry out measurement within a calculations that involve various measurement system (e.g., 4’3” units of measurement. plus 7’10” equals 12”1”). 4.2.D.2 – Use approximate equivalents between standard and metric systems to estimate measurements (e.g., 5 kilometers is about 3 miles). 4.2.D.3 – Recognize that the Pg. 506, No. 8: Approximate degree of precision needed in quantities with increasing calculations depends on how the degrees of accuracy. results will be used and the instruments used to generate the measurements. 4.2.D.4 – Select and use Pg. 302, No. 9: Determine the appropriate units and tools to degree of accuracy needed in a measure quantities to the degree given situation and choose units of precision needed in a accordingly. particular problem-solving situation. 171 4.2.D.5 – Recognize that all Pg. 302, No. 10: Understand that measurements of continuous all measurements of continuous quantities are approximations. quantities are approximate. 4.2.D.6 – Solve problems that involve compound measurement units, such as speed (miles per hour), air pressure (pounds per square inch), and population density (persons per square mile). 4.2.E. Measuring Geometric Objects 4.2.E.1 – Develop and apply Pg. 238, No. 16: Develop, strategies for finding perimeter understand, and apply a variety and area. of strategies for determining • Geometric figures made perimeter, area, surface area, by combining triangles, angle measure, and volume. rectangles and circles or Pg. 302, No. 7: Use estimated parts of circles and actual measurements to • Estimation of area using describe and compare grids of various sizes phenomena. • Impact of a dilation on the perimeter and area of a 2-dimensional figure 4.2.E.2 – Recognize that the volume of a pyramid or cone is one-third of the volume of the 172 prism or cylinder with the same base and height (e.g., use rice to compare volumes of figures with the same base and height. 4.2.E.3 – Develop and apply Pg. 303, No. 11: Develop strategies and formulas for formulas and procedures for finding the surface area and solving problems related to volume of a three-dimensional measurement. figure. Pg. 507, No. 10: Develop • Volume – prism, cone, informal ways of approximating pyramid the surface area and volume of • Surface area – prism familiar objects, and discuss (triangular or rectangular whether the approximations base), pyramid (triangular make sense. or rectangular base) • Impact of a dilation on Pg. 508, No. 11: Express the surface area and mathematically and explain the volume of a three- impact of the change of an dimensional figure object’s linear dimensions on its surface area and volume. 4.2.E.4 – Use formulas to find the volume and surface area of a sphere. 4.3 All students will represent 4.3.A. Patterns Pg. 357, Overview and analyze relationships among variable quantities and 4.3.A.1 – Recognize, describe, Pg. 358, No. 8: Understand and solve problems involving extend, and create patterns describe the relationships among patterns, functions, and involving whole numbers, various representations of algebraic concepts and rational numbers, and integers. patterns and functions. 173 processes. • Descriptions using tables, Pg. 202, No. 16: Recognize and 4.3.A. Patterns verbal and symbolic rules, describe patterns in both finite 4.3.B. Functions and graphs, simple equations and infinite number sequences Relationships or expressions involving whole numbers, 4.3.C. Modeling • Finite and infinite rational numbers, and integers. 4.3.D. Procedures sequences Pg. 504, No. 5: Develop an • Arithmetic sequences understanding of infinite (i.e., sequences generated sequences that arise in natural by repeated addition of a situations. fixed number, positive or Pg. 362, No. 13: Develop, negative) analyze, and explain arithmetic • Geometric sequences sequences. (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1) • Generating sequences by using calculators to repeatedly apply a formula 4.3.B. Functions and Pgs. 427-428, Overview Relationships 4.3.B.1 – Graph functions, and Pg. 360, No. 11: Understand and understand and describe their describe the general behavior of general behavior. functions. • Equations involving two Pg. 430, No. 8: Analyze tables variables and graphs to identify properties and relationships. • Rates of change (informal notion of slope) 174 4.3.B.2 – Recognize and describe Pg. 430, No. 8: Analyze tables the difference between linear and and graphs to identify properties exponential growth, using tables, and relationships. graphs, and equations. 4.3.C. Modeling 4.3.C.1 – Analyze functional Pg. 304, No. 15: Understand and relationships to explain how a explain the impact of the change change in one quantity can result of an object’s linear dimensions in a change in another, using on its perimeter, area, or volume. pictures, graphs, charts, and Pg. 360, No. 10: Analyze equations. functional relationships to explain how a change in one quantity results in a change in another. 4.3.C.2 – Use patterns, relations, Pg. 358, No. 7: Represent and symbolic algebra, and linear describe mathematical functions to model situations. relationships with tables, rules, • Using concrete materials simple equations, and graphs. (manipulatives), tables, Pg. 361, No. 12: Use patterns, graphs, verbal rules, relationships, and linear functions algebraic to model situations in expressions/equations/in- mathematics and in other areas. equalities Pg. 429, No. 6: Represent • Growth situations, such as situations and number patterns population growth and with concrete materials, tables, compound interest, using graphs, verbal rules, and standard recursive (e.g., NOW- algebraic notation. NEXT) formulas (cf. Pg. 504, No. 4: Recognize and science standard 5.5 and express the difference between 175 social studies standard linear and exponential growth. 6.6) 4.3.D. Procedures 4.3.D.1 – Use graphing Pg. 430, No. 7: Use graphing techniques on a number line. techniques on a number line to • Absolute value model both absolute value and • Arithmetic operations arithmetic operations. represented by vectors Pg. 432, No. 12: Investigate (arrows) (e.g., “-3 + 6” is inequalities and nonlinear “left 3, right 6”) equations informally. 4.3.D.2 – Solve simple linear Pg. 432, No. 10: Solve simple equations informally, graphically, linear equations using concrete, and using formal algebraic informal, and graphical methods, methods. as well as appropriate paper-and- • Multi-step, integer pencil techniques. coefficients only Pg. 432, No. 11: Explore linear (although answers may equations through the use of not be integers) calculators, computers, and other • Using paper-and-pencil, technology. calculators, graphing calculators, spreadsheets, and other technology • Simple literal equations (e.g., A = lw) 4.3.D.3 – Solve simple linear Pg. 432, No. 12: Investigate inequalities. inequalities and nonlinear equations informally. 176 4.3.D.4 – Create, evaluate, and simplify algebraic expressions involving variables. • Order of operations, including appropriate use of parentheses • Distributive property • Substitution of a number for a variable • Translation of a verbal phrase or sentence into an algebraic expression, equation, or inequality, and vice versa 4.3.D.5 – Understand and apply Pg. 432, No. 12: Investigate the properties of operations, inequalities and nonlinear numbers, equations, and equations informally. inequalities. • Additive inverse • Multiplicative inverse • Addition and multiplication properties of equality • Addition and multiplication properties of inequalities 4.4 All students will develop an 4.4.A. Data Analysis understanding of the concepts and techniques of data 4.4.A.1 – Select and use Pg. 394, No. 9: Generate, analysis, probability, and appropriate representations for collect, organize, and analyze 177 discrete mathematics, and sets of data, and measures of data and represent this data in will use them to model central tendency (mean, median, tables, charts, and graphs. situations, solve problems, and mode). Pg. 394, No. 10: Select and use and analyze and draw • Type of display most appropriate graphical appropriate inferences from appropriate for given data representations and measures of data. • Box-and-whisker plot, central tendency (mean, mode, 4.4.A. Data Analysis upper quartile, lower and median) for sets of data. 4.4.B. Probability quartile 4.4.C. Discrete Mathematics – • Scatter plot Systematic Listing and Counting • Calculators and computer 4.4.D. Discrete Mathematics – used to record and Vertex-Edge Graphs and process information Algorithms • Finding the median and mean (weighted average) using frequency data • Effect of additional data on measures of central tendency 4.4.A.2 – Make inferences and Pg. 395, No. 11: Make formulate and evaluate inferences and formulate and arguments based on displays and evaluate arguments based on data analysis of data sets. analysis and data displays. 4.4.A.3 – Estimate lines of best Pg. 395, No. 12: Use lines of fit and use them to interpolate best fit to interpolate and predict within the range of the data. from data. 4.4.A.4 – Use surveys and sampling techniques to generate data and draw conclusions about large groups. 178 4.4.B. Probability Pgs. 392-393, Overview 4.4.B.1 – Interpret probabilities Pg. 396, No. 16: Interpret as ratios, percents, and decimals. probabilities as ratios and percents. 4.4.B.2 – Determine probabilities Pg. 396, No. 13: Determine the of compound events. probability of a compound event. 4.4.B.3 – Explore the probabilities of conditional events (e.g., if there are seven marbles in a bag, three red and four green, what is the probability that two marbles picked from the bag, without replacement, are both red). 4.4.B.4 – Model situations Pg. 396, No. 15: Use models of involving probability with probability to predict events simulations (using spinners, dice, based on actual data. calculators and computers) and theoretical models. • Frequency, relative frequency 4.4.B.5 – Estimate probabilities Pg. 396, No. 14: Model and make predictions based on situations involving probability, experimental and theoretical such as genetics, using both probabilities. simulations and theoretical methods. 179 4.4.B.6 – Play and analyze probability-based games, and discuss the concepts of fairness and expected value. 4.4.C. Discrete Mathematics – Pgs. 471-472, Overview Systematic Listing and Counting 4.4.C.1 – Apply the Pg. 473, No. 6: Use systematic multiplication principle of listing, counting, and reasoning counting. in a variety of different contexts. • Permutations: ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23-student class) • Factorial notation • Concept of combinations (e.g., number of possible delegations of 3 out of 23 students) 4.4.C.2 – Explore counting Pg. 476, No. 9: Explore methods problems involving Venn for storing, processing, and diagrams with three attributes communicating information. (e.g., there are 15, 20, and 25 180 students respectively in the chess club, the debating team, and the engineering society; how many different students belong to the three clubs if there are 6 students in chess and debating, 7 students in chess and engineering, 8 students in debating and engineering, and 2 students in all three?). 4.4.C.3 – Apply techniques of Pg. 359, No. 9: Use patterns, systematic listing, counting, and relationships, and functions to reasoning in a variety of different model situations and to solve contexts. problems in mathematics and in other subject areas. 4.4.D. Discrete Mathematics – Vertex-Edge Graphs and Algorithms 4.4.D.1 – Use vertex-edge graphs Pg. 474, No. 7: Recognize and algorithmic thinking to common discrete mathematical represent and find solutions to models, explore their properties, practical problems. and design them for specific • Finding the shortest situations. network connecting Pg. 477, No. 10: Devise, specified sites describe, and test algorithms for • Finding a minimal route solving optimization and search that includes every street problems. (e.g., for trash pick-up) 181 • Finding the shortest route on a map from one site to another • Finding the shortest circuit on a map that makes a tour of specified sites • Limitations of computers (e.g., the number of routes for a delivery truck visiting n sites is n!, so finding the shortest circuit by examining all circuits would overwhelm the capacity of any computer, now or in the future, even if n is less than 100) 182 STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATIONS, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS. Descriptive Statement: The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards. Problem Solving: Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas. Communication: Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress. Connections: Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life. Reasoning: Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied. 183 Representations: Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics. Technology: Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper- and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems). 184 Mathematics 4.5 Process Standard At each grade level, with respect to content appropriate for that grade level, students will: A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). • Open-ended problems • Non-routine problems • Problems with multiple solutions • Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., “try a simpler problem” or “make a diagram”) to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. 6. Distinguish relevant from irrelevant information, and identify missing information. B. Communication 1. Use communication to organize and clarify their mathematical thinking. • Reading and writing • Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 185 D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. • Counterexamples as a means of disproving conjectures • Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. • Concrete representations (e.g., base-ten blocks or algebra tiles) • Pictorial representations (e.g., diagrams, charts, or tables) • Symbolic representations (e.g., a formula) • Graphical representations (e.g., a line graph) 2. Select, apply and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information. 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators and problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based technology for mathematical applications in the sciences (cf. science standards). 186 New Jersey Core Curriculum Content Standards for Mathematics (Text underlined reflects changes adopted on January 9, 2008) (Arranged by Strand across All Grade Levels, and including Preschool Learning Expectations in Mathematics) New Jersey Department of Education Office of Academic Standards January 2008 187 Revisions to New Jersey’s Mathematics Standards as adopted on January 9, 2008 The newly adopted changes may be found on pages 11, 17, 19, 23, 27, 31, 33, 40, and 45. Additions have been underlined and deletions have been bracketed. On page 11, the word “rates” is added to 4.1.8A3; and a new bullet, “Least common multiple, greatest common factor,” is added to clarify 4.1.6A7. On page 17, three new cumulative progress indicators (CPIs) are added. The new 4.2.8A7 specifies that students will “Create two-dimensional representations (e.g., nets or projective views) for the surfaces of three-dimensional objects.” The new 4.2.8A6 and 4.2.12A5 add the content that students will “Perform basic geometric constructions using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology.” On page 19, 4.2.12B1 and 4.2.12C1 are edited slightly for clarification. The new 4.2.12C3 adds the content that students will be able to “Find an equation of a circle given its center and radius and, given an equation of a circle in standard form, find its center and radius.” On page 23, a new bullet, “Special right triangles,” is added to 4.2.12E1. On page 27, a new bullet, “Solutions of systems of linear inequalities using graphing techniques,” is added to 4.3.12B2; also the original portion of the CPI is edited to explicitly include both algebraic and graphing techniques. The seventh bullet is shortened to “Solutions of systems of equations.” Combined with the inclusion of both algebraic and graphing techniques, the intent is to now include the algebraic solution of systems of equations. In the first bullet of this CPI, the slope of a “curve” is deleted. On page 31, three CPIs are edited. A new bullet, “literal equations,” is added to both 4.3.8D2 and 4.3.12D2. CPI 4.3.12D2 is further edited to clarify the inclusion of inequalities. Two new bullets, (“Perform simple operations with rational expressions” and “Evaluate polynomial and rational expressions”) are added to 4.2.12D1. On page 33, 4.4.8A2 is edited to clarify the inclusion of data “sets.” A new bullet, “Correlation vs. causation,” is added to 4.4.12A2. The new 4.4.12A6 adds the content that students will “Distinguish between randomized experiments and observational studies.” On page 40, the new 4.5A6 adds the content that at each grade level, with respect to content appropriate for that grade level, students will “Distinguish relevant from irrelevant information, and identify missing information.” On page 45, the 2004 Achieve, Inc. document, “Ready or Not: Creating a High School Diploma That Counts,” is added to the list of references. This document is available online at http://www.achieve.org/publications/national_reports_view and includes the American Diploma Project Benchmarks in Mathematics. Additional information concerning New Jersey’s participation in the American Diploma Project may be found at www.njhighschoolsummit.org/events.asp (Inside front cover) Adopted January 9, 2008 188 Adopted January 9, 2008 PREFACE This document is a newly formatted version of the New Jersey Core Curriculum Content Standards for Mathematics, as revised and adopted by the New Jersey State Board of Education in July 2002 and revised in January 2008. It was developed in response to requests from schools and school districts for a version that would make it easier to track the learning of specific mathematics content across grade levels. The mathematics content and numbering of the cumulative progress indicators in this version of the standards remain unchanged from the version adopted by the State Board of Education in July 2002. Consequently, in order to align related content across grades, the indicators within a particular grade level have sometimes been arranged out of numerical order. The descriptive statements accompanying each of the five standards have been broken up into pieces, each of which now accompanies the lettered strand to which it refers. In all cases, however, it is the formatting and arrangement that are new; the content remains unchanged. It is also worth emphasizing that the goal remains unchanged: To enable ALL of New Jersey’s children to acquire the mathematical skills, understandings, and attitudes that they will need to be successful in their careers and daily lives. The New Jersey Core Curriculum Content Standards are intended for all students. This includes students who are college-bound or career-bound, gifted and talented, those whose native language is not English, students with disabilities, and students from diverse socioeconomic backgrounds. State Board adoption of the revised Core Curriculum Content Standards for Mathematics means that every student will be involved in experiences addressing all of the expectations set forth in the standards. It does not mean that all students will be enrolled in the same courses. Different groups of students should address the standards at different levels of depth and may complete the core curriculum according to different timetables. Depending on their interests, abilities, and career plans, many students will and should develop knowledge and skills that go beyond the specific indicators of the Core Curriculum Content Standards. Finally, the answers to a series of frequently asked questions concerning the revised standards are available on the Department’s website. For the convenience of those receiving this document, the questions and answers have been reprinted here, following the content of the last mathematics standard. For additional information, including suggested teaching strategies for implementing these standards, and for sample assessment items linked to the Statewide assessments, educators are encouraged to explore the Department’s website, at http://www.state.nj.us/education/. Adopted January 9, 2008 189 Adopted January 9, 2008 STANDARD 4.1 (NUMBER AND NUMERICAL OPERATIONS) ALL STUDENTS WILL DEVELOP NUMBER SENSE AND WILL PERFORM STANDARD NUMERICAL OPERATIONS AND ESTIMATIONS ON ALL TYPES OF NUMBERS IN A VARIETY OF WAYS. Number Sense. Number sense is an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent that comes from investigating their characteristics and using them in diverse situations. It involves an understanding of how different types of numbers, such as fractions and decimals, are related to each other, and how each can best be used to describe a particular situation. It subsumes the more traditional category of school mathematics curriculum called numeration and thus includes the important concepts of place value, number base, magnitude, and approximation and estimation. Preschool Learning 4.1.2A. Number Sense 4.1.3 A. Number Sense 4.1.4 A. Number Sense 4.1.5 A. Number Sense Expectations Grade 2 Grade 3 Grade 4 Grade 5 EXPECTATION 1: By the end of Grade 2, Building upon knowledge and Building upon knowledge and Building upon knowledge and Children students will: skills gained in preceding skills gained in preceding skills gained in preceding demonstrate an grades, by the end of Grade 3, grades, by the end of Grade 4, grades, by the end of Grade 5, understanding of number students will: students will: students will: and numerical operations. 1.1 Demonstrates understanding of 1. Use real-life experiences, 1. Use real-life experiences, 1. Use real-life experiences, 1. Use real-life experiences, one-to-one correspondence (e.g., physical materials, and physical materials, and physical materials, and physical materials, and places one placemat at each place, gives each child one cookie, places technology to construct technology to construct technology to construct technology to construct one animal in each truck, hands out meanings for numbers meanings for numbers meanings for numbers meanings for numbers manipulatives to be shared with a (unless otherwise noted, all (unless otherwise noted, all (unless otherwise noted, all (unless otherwise noted, all friend saying "One for you, one for me."). indicators for grade 2 pertain to indicators for grade 3 pertain to indicators for grade 4 pertain to indicators for grade 5 pertain to these sets of numbers as well). these sets of numbers as well). these sets of numbers as well). these sets of numbers as well). 1.3 Learns to say the • Whole numbers • Whole numbers through • Whole numbers through [Exploration of negative numbers is counting numbers. through hundreds hundred thousands millions included in 4.1.4 A 7 below.] • Ordinals • Commonly used fractions • Commonly used fractions • All fractions as part of a whole, • Proper fractions (denominators of 2, 3, 4, 5, (denominators of 2, 3, 4, 5, 6, 8, as subset of a set, as a location (denominators of 2, 3, 6, 8, 10) as part of a whole, 10, 12, and 16) as part of a on a number line, and as as a subset of a set, and as a whole, as a subset of a set, and divisions of whole numbers 4, 8, 10) location on a number line as a location on a number line 4. Explore the extension of the place value • Decimals through hundredths • All decimals system to decimals through hundredths. 1.5 Recognizes and names 2. Demonstrate an under- 2. Demonstrate an under- 2. Demonstrate an under- [Use of concrete representations some written numerals. standing of whole number standing of whole number standing of place value (e.g., base-ten blocks) is included in place value concepts. place value concepts. concepts. indicator 4.5 E 1.] 3. Identify whether any whole 3. Demonstrate a sense of the 3. Demonstrate a sense of the number is odd or even. relative magnitudes of numbers. relative magnitudes of numbers. 1.4 Discriminates numbers from [Recognizing orders of magnitude associated with large and small other symbols in the environment physical quantities is included in science indicator 5.3.4 A 2.] (e.g., street signs, license plates, room number, clock, etc.). 3. Understand that numbers 5. Understand the various 4. Understand the various [According to Preschool Health, Safety have a variety of uses. uses of numbers. uses of numbers. and Physical Education Expectation 3.5 Knows how to dial 911 for help.] 4. Count and perform simple • Counting, measuring, • Counting, measuring, 1.2 Spontaneously counts for computations with coins. labeling (e.g., numbers on labeling (e.g., numbers on own purposes (e.g., counting baseball uniforms) blocks or cars, counting baseball uniforms), beads while stringing them, • Amounts up to $1.00 [Counting money is also locating (e.g., Room 235 2. Recognize the decimal nature (using cents notation) included in indicators is on the second floor) of United States currency and handing out napkins). 4.1.3 B 5 and 4.1.4 B 6.] compute with money. 5. Use concrete and pictorial 4. Use whole numbers, models to relate whole numbers, fractions, and decimals to commonly used fractions, and represent equivalent forms decimals to each other, and to represent equivalent forms of the of the same number. same number. 5. Develop and apply number theory concepts in problem solving situations. • Primes, factors, multiples 1.6 Compares numbers in 5. Compare and order 6. Compare and order 6. Compare and order 6. Compare and order different contexts whole numbers. numbers. numbers. numbers. (e.g., using words such as 7. Explore settings that give more and less). [Use of integers is included in Adopted January 9, 2008 190 Adopted January 9, 2008 rise to negative numbers. science indicator 5.3.4 A 3.] • Temperatures below 0o, debts • Extension of the number line Adopted January 9, 2008 191 Adopted January 9, 2008 4.1 NUMBER AND NUMERICAL OPERATIONS Descriptive Statement: Numbers and arithmetic operations are what most of the general public think about when they think of mathematics; and, even though other areas like geometry, algebra, and data analysis have become increasingly important in recent years, numbers and operations remain at the heart of mathematical teaching and learning. Facility with numbers, the ability to choose the appropriate types of numbers and the appropriate operations for a given situation, and the ability to perform those operations as well as to estimate their results, are all skills that are essential for modern day life. 4.1.6 A. Number Sense 4.1.7 A. Number Sense 4.1.8 A. Number Sense 4.1.12 A. Number Sense Grade 6 Grade 7 Grade 8 Grade 12 Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the end of Grade 6, students will: end of Grade 7, students will: end of Grade 8, students will: end of Grade 12, students will: 1. Use real-life experiences, 1. Extend understanding of the 1. Extend understanding of the 1. Extend understanding of the physical materials, and number system by number system by number system to all real technology to construct constructing meanings for the constructing meanings for the numbers. meanings for numbers (unless following (unless otherwise following (unless otherwise otherwise noted, all indicators noted, all indicators for grade 7 noted, all indicators for grade 8 for grade 6 pertain to these sets pertain to these sets of pertain to these sets of of numbers as well). numbers as well): numbers as well): • All integers • All fractions as part of a • Rational numbers • Rational numbers whole, as subset of a set, as a • Percents • Percents location on a number line, and • Exponents as divisions of whole numbers • Roots • Absolute values • All decimals • Whole numbers with exponents • Numbers represented in scientific notation 3. Demonstrate a sense of the 2. Demonstrate a sense of the 2. Demonstrate a sense of the relative magnitudes of numbers. relative magnitudes of numbers. relative magnitudes of numbers. 6. Understand that all fractions 6. Recognize that repeating decimals correspond to can be represented as repeating fractions and determine their fractional equivalents. or terminating decimals. • 5/7 = 0. 714285714285… = 0. 714285 4. Explore the use of ratios and 3. Understand and use ratios, 3. Understand and use ratios, proportions in a variety of proportions, and percents rates, proportions, and situations. (including percents greater percents (including percents 5. Understand and use whole- than 100 and less than 1) in a greater than 100 and less than number percents between 1 and variety of situations. 1) in a variety of situations. 100 in a variety of situations. 2. Recognize the decimal nature of United States currency and compute with money. 6. Use whole numbers, 5. Use whole numbers, 5. Use whole numbers, [Relate to indicator 4.5 E 2, fractions, and decimals to fractions, decimals, and fractions, decimals, and select, apply, and translate represent equivalent forms of percents to represent percents to represent among mathematical the same number. equivalent forms of the same equivalent forms of the same representations to solve number. number. problems.] 7. Develop and apply number theory concepts in problem solving situations. • Primes, factors, multiples • Common multiples, common factors • Least common multiple, greatest common factor 8. Compare and order numbers. 4. Compare and order numbers 4. Compare and order numbers 2. Compare and order rational of all named types. of all named types. and irrational numbers. [Use of graphing techniques 7. Construct meanings for 3. Develop conjectures and on a number line is included common irrational numbers, informal proofs of properties in indicator 4.3.7 D 1.] such as π (pi) and the square of number systems and sets root of 2. of numbers. Adopted January 9, 2008 192 Adopted January 9, 2008 Numerical Operations. Numerical operations are an essential part of the mathematics curriculum, especially in the elementary grades. Students must be able to select and apply various computational methods, including mental math, pencil-and- paper techniques, and the use of calculators. Students must understand how to add, subtract, multiply, and divide whole numbers, fractions, decimals, and other kinds of numbers. With the availability of calculators that perform these operations quickly and accurately, the instructional emphasis now is on understanding the meanings and uses of these operations, and on estimation and mental skills, rather than solely on the development of paper-and-pencil proficiency. Preschool Learning 4.1.2 B. Numerical 4.1.3 B. Numerical 4.1.4 B. Numerical 4.1.5 B. Numerical Expectations Operations Grade 2 Operations Grade 3 Operations Grade 4 Operations Grade 5 1.8 Adds two groups of 1. Develop the meanings of 1. Develop the meanings of 1. Develop the meanings of 1. Recognize the concrete objects by counting addition and subtraction the four basic arithmetic the four basic arithmetic appropriate use of each the total (e.g., three blue pegs, by concretely modeling operations by modeling operations by modeling arithmetic operation in three yellow pegs, six pegs and discussing a large and discussing a large and discussing a large problem situations. altogether). variety of problems. variety of problems. variety of problems. • Joining, separating, • Addition and • Addition and 1.9 Subtracts one group of concrete objects from and comparing subtraction: joining, subtraction: joining, another by taking some separating, comparing separating, comparing away and then counting the 2. Explore the meanings of • Multiplication: • Multiplication: repeated remainder (e.g., "I have four multiplication and repeated addition, addition, area/array carrot sticks. I'm eating one! division by modeling and area/array • Division: repeated Now I have 3!"). discussing problems. • Division: repeated subtraction, sharing subtraction, sharing 3. Develop proficiency with 2. Develop proficiency with 2. Develop proficiency with basic addition and subtraction basic multiplication and basic multiplication and number facts using a variety division number facts division number facts using a of fact strategies (such as using a variety of fact variety of fact strategies (such “counting on” and “near strategies (such as “skip as “skip counting” and doubles”) and then commit counting” and “repeated “repeated subtraction”) and them to memory. subtraction”). then commit them to memory. [The Foundations for 4. Construct, use, and explain 3. Construct, use, and explain 3. Construct, use, and explain 2. Construct, use, and explain performing addition and procedures for performing procedures for performing procedures for performing procedures for performing subtraction calculations addition and subtraction whole number calculations whole number calculations addition and subtraction with are laid through activities calculations with: with: and with: fractions and decimals with: associated with Preschool • Pencil-and-paper • Pencil-and-paper • Pencil-and-paper • Pencil-and-paper Mathematics Expectations • Mental math • Mental math • Mental math • Mental math 1.8 and 1.9 above] • Calculator • Calculator • Calculator • Calculator 5. Use efficient and accurate 4. Use efficient and accurate 4. Use efficient and accurate 3. Use an efficient and accurate pencil-and-paper procedures pencil-and-paper procedures pencil-and-paper procedures pencil-and-paper procedure for computation with whole for computation with whole for computation with for division of a 3-digit numbers. numbers. whole numbers. number by a 2-digit number. • Addition of 2-digit numbers • Addition of 3-digit numbers • Addition of 3-digit numbers • Subtraction of 2-digit • Subtraction of 3-digit • Subtraction of 3-digit numbers numbers numbers • Multiplication of 2-digit • Multiplication of 2-digit numbers by 1-digit numbers numbers • Division of 3-digit numbers by 1-digit numbers 5. Construct and use [Explaining procedures for procedures for performing decimal addition performing decimal and subtraction is included addition and subtraction. in 4.1.5 B 2 above.] 5. Count and perform simple 6. Count and perform simple [Counting coins up to $1.00 computations with money. computations with money. (cents notation) is included in indicator 4.1.2 A 4.] • Cents notation (¢) • Standard dollars and cents notation 6. Select pencil-and-paper, 6. Select pencil-and-paper, 7. Select pencil-and-paper, 4. Select pencil-and-paper, mental math, or a calculator mental math, or a calculator mental math, or a calculator mental math, or a calculator as the appropriate as the appropriate as the appropriate as the appropriate computational method in a computational method in a computational method in a computational method in a given situation depending given situation depending given situation depending given situation depending on the context and numbers. on the context and numbers. on the context and numbers. on the context and numbers. 4.1 Strand B, Numerical Operations, is continued on the next page Adopted January 9, 2008 193 Adopted January 9, 2008 4.1 NUMBER AND NUMERICAL OPERATIONS 4.1.6 B. Numerical Operations 4.1.7 B. Numerical Operations 4.1.8 B. Numerical Operations 4.1.12 B. Numerical Operations Grade 6 Grade 7 Grade 8 Grade 12 1. Recognize the appropriate use of each arithmetic operation [Applying mathematics in practical situations and in in problem situations. other disciplines is included in indicator 4.5 C 4.] 1. Use and explain procedures for performing calculations involving addition, 2. Construct, use, and explain 1. Use and explain procedures subtraction, multiplication, 1. Extend understanding and use procedures for performing for per-forming calculations division, and exponentiation of operations to real numbers calculations with fractions with integers and all number with integers and all number and algebraic procedures. and decimals with: types named above with: types named above with: • Pencil-and-paper • Pencil-and-paper • Pencil-and-paper • Mental math • Mental math • Mental math • Calculator • Calculator • Calculator 3. Use an efficient and accurate . pencil-and-paper procedure for division of a 3-digit number by a 2-digit number. [Procedures for performing decimal multiplication and division are included in 4.1.6 B 2 above.] [Compound interest is included in indicators 4.3.7 C 1, 4.3.8 C 2, and 4.3.12 C 1.] 4. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers. 4.1 Strand B, Numerical Operations, is continued on the next page Adopted January 9, 2008 194 Adopted January 9, 2008 4.1.2 B. Numerical Operations 4.1.3 B. Numerical Operations 4.1.4 B. Numerical Operations 4.1.5 B. Numerical Operations Grade 2 (continued) Grade 3 (continued) Grade 4 (continued) Grade 5 (continued) No Associated Preschool Learning Expectations 7. Check the reasonableness of 7. Check the reasonableness of 8. Check the reasonableness of 5. Check the reasonableness of results of computations. results of computations. results of computations. results of computations. 9. Use concrete models to [Formal procedures for adding explore addition and and subtracting fractions are subtraction with fractions. included in 4.1.5 B 2 above.] 8. Understand and use the 10. Understand and use the 6. Understand and use the inverse relationship between inverse relationships between various relationships among addition and subtraction. addition and subtraction and operations and properties of operations. between multiplication and division. Estimation. Estimation is a process that is used constantly by mathematically capable adults, and one that can be easily mastered by children. It involves an educated guess about a quantity or an intelligent prediction of the outcome of a computation. The growing use of calculators makes it more important than ever that students know when a computed answer is reasonable; the best way to make that determination is through the use of strong estimation skills. Equally important is an awareness of the many situations in which an approximate answer is as good as, or even preferable to, an exact one. Students can learn to make these judgments and use mathematics more powerfully as a result. Preschool Learning 4.1.2 C. Estimation 4.1.3 C. Estimation 4.1.4 C. Estimation 4.1.5 C. Estimation Expectations Grade 2 Grade 3 Grade 4 Grade 5 1. Judge without counting 1. Judge without counting 1. Judge without counting whether a set of objects whether a set of objects whether a set of objects has less than, more than, has less than, more than, has less than, more than, or the same number of or the same number of or the same number of objects as a reference set. objects as a reference set. objects as a reference set. 1.7 Uses estimation as a 3. Explore a variety of 2. Construct and use a variety 2. Construct and use a variety 1. Use a variety of method for approximating strategies for estimating of estimation strategies of estimation strategies estimation strategies for an appropriate amount both quantities (e.g., the (e.g., rounding and mental (e.g., rounding and mental both number and (e.g., at snack time, number of marbles in a math) for estimating both math) for estimating both computation. deciding how many jar) and results of quantities and the result of quantities and the results of napkins to take from a computation. computations. computations. large pile for the group, 3. Recognize when an 3. Recognize when an 2. Recognize when an determining number of estimate is appropriate, estimate is appropriate, estimate is appropriate, blocks to use when and understand the and understand the and understand the building structures). usefulness of an estimate usefulness of an estimate usefulness of an as distinct from an exact as distinct from an exact estimate as distinct from answer. answer. an exact answer. 2. Determine the 4. Use estimation to 4. Use estimation to 3. Determine the reasonableness of an determine whether the determine whether the reasonableness of an answer by estimating the result of a computation result of a computation answer by estimating the result of computations (either by calculator or (either by calculator or result of operations. (e.g., 15 + 16 is not 211). by hand) is reasonable. by hand) is reasonable. [Relate to science indicator 5.3.4 A 1, determining 4. Determine whether a given the reasonableness of estimates, measurements, and estimate is an overestimate computations when doing science.] or an underestimate. Adopted January 9, 2008 195 Adopted January 9, 2008 4.1 NUMBER AND NUMERICAL OPERATIONS 4.1.6 B. Numerical Operations 4.1.7 B. Numerical Operations 4.1.8 B. Numerical Operations 4.1.12 B. Numerical Operations Grade 6 (continued) Grade 7 (continued) Grade 8 (continued) Grade 12 (continued) 2. Use exponentiation to find 2. Use exponentiation to find 2. Develop, apply, and explain whole number powers of whole number powers of methods for solving problems numbers. numbers. involving rational and negative exponents. 5. Find squares and cubes of 3. Find square and cube roots of 4. Understand and apply the whole numbers. numbers and understand the laws of exponents to simplify inverse nature of powers and expressions involving roots. numbers raised to powers. 6. Check the reasonableness of [Relate to Science Indicator 5.3.4 A 1, determining the reasonableness results of computations. of estimates, measurements, and computations when doing science.] 4. Solve problems involving proportions and percents. 7. Understand and use the various relationships among operations and properties of operations. 8. Understand and apply the 3. Understand and apply the 5. Understand and apply the standard algebraic order of standard algebraic order of standard algebraic order of operations for the four basic operations, including operations, including operations, including appropriate use of appropriate use of appropriate use of parentheses. parentheses. parentheses. 3. Perform operations on matrices. • Addition and subtraction • Scalar multiplication 4.1.6 C. Estimation 4.1.7 C. Estimation 4.1.8 C. Estimation 4.1.12 C. Estimation Grade 6 Grade 7 Grade 8 Grade 12 1. Estimate square and cube roots of numbers. 1. Use a variety of strategies for 1. Use equivalent 2. Use equivalent estimating both quantities representations of numbers representations of numbers and the results of such as fractions, decimals, such as fractions, decimals, computations. and percents to facilitate and percents to facilitate estimation. estimation. 2. Recognize when an estimate 3. Recognize the limitations of 1. Recognize the limitations of is appropriate, and estimation and assess the estimation, assess the amount understand the usefulness of amount of error resulting of error resulting from an estimate as distinct from from estimation. estimation, and determine an exact answer. whether the error is within acceptable tolerance limits. 3. Determine the reasonableness of an answer by estimating [Relate to indicator 4.5 D 4, relying on reasoning, rather than the result of operations. answer keys, to check the correctness of problem solutions.] 4. Determine whether a given estimate is an overestimate or an underestimate. Adopted January 9, 2008 196 Adopted January 9, 2008 STANDARD 4.2 (GEOMETRY AND MEASUREMENT) ALL STUDENTS WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES, RELATIONSHIPS, AND MEASUREMENT TO MODEL, DESCRIBE, AND ANALYZE PHENOMENA. Geometric Properties. This includes identifying, describing and classifying standard geometric objects, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity. Preschool Learning 4.2.2 A. Geometric 4.2.3 A. Geometric 4.2.4 A. Geometric 4.2.5 A. Geometric Expectations Properties Grade 2 Properties Grade 3 Properties Grade 4 Properties Grade 5 EXPECTATION 2: By the end of Grade 2, Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills students will: gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the Children develop knowledge end of Grade 3, students will: end of Grade 4, students will: end of Grade 5, students will: of spatial concepts, e.g., 1. Identify and describe spa- 1. Identify and describe 1. Identify and describe shapes and measurement. tial relationships among spatial relationships of two spatial relationships of two objects in space and their or more objects in space. or more objects in space. relative shapes and sizes. • Direction, orientation, • Direction, orientation, and 2.5 Uses positional words • Inside/outside, left/right, and perspectives (e.g., perspectives (e.g., which in a functional way above/below, between object is on your left when which object is on your you are standing here?) (e.g., "I put the red block on • Smaller/larger/same size, left when you are top of the cabinet."). wider/ narrower, longer/shorter • Relative shapes and sizes standing here?) • Shadows (projections) of • Congruence (i.e., same size and shape) • Relative shapes and sizes everyday objects 2.1 Identifies basic shapes in 2. Use concrete objects, drawings, 2. Use properties of standard 2. Use properties of standard 2. Identify, describe, compare, the environment (e.g., circle, and computer graphics to three-dimensional and three-dimensional and two- and classify polygons. square, triangle, cube, sphere). identify, classify, and describe two-dimensional shapes dimensional shapes to identify, • Triangles by angles and sides standard three-dimensional classify, and describe them. • Quadrilaterals, including to identify, classify, and 2.6 Makes three-dimensional and two-dimensional shapes. • Vertex, edge, face, side, angle squares, rectangles, parallelo- describe them. constructions and models • Vertex, edge, face, side • 3D figures – cube, rectangular grams, trapezoids, rhombi • Vertex, edge, face, side, prism, sphere, cone, cylinder, (e.g., sculptures that have • 3D figures – cube, rectangular • Polygons by number of sides. angle and pyramid height, depth, and width). prism, sphere, cone, cylinder, • Equilateral, equiangular, • 3D figures – cube, • 2D figures – square, rectangle, and pyramid regular rectangular prism, sphere, circle, triangle, quadrilateral, 2.7 Makes connections • 2D figures – square, rectangle, • All points equidistant from circle, triangle cone, cylinder, and pentagon, hexagon, octagon between two-dimensional and a given point form a circle three-dimensional forms • Relationships between three- and pyramid • Inclusive relationships – • 2D figures – square, squares are rectangles, cubes (e.g., circle-sphere, square- two-dimensional shapes (i.e., the are rectangular prisms cube, triangle-pyramid). face of a 3D shape is a 2D shape) rectangle, circle, triangle, pentagon, hexagon, octagon [Models of 3D objects are 3. Identify similar figures. included in Preschool Mathematics Expectation 2.6 above] [Identifying basic shapes 3. Describe, identify and 3. Identify and describe relationships 3. Identify and describe relationships 4. Understand and apply the in the environment is create instances of line among two-dimensional shapes. among two-dimensional shapes. concepts of congruence and included in Preschool symmetry. • Same size, same shape • Congruence symmetry (line and rotational). • Lines of symmetry • Lines of symmetry Mathematics Expectation 2.1 above] 4. Understand and apply 4. Understand and apply 1. Understand and apply concepts concepts involving lines, concepts involving lines, involving lines and angles. angles, and circles. angles, and circles. •Notation for line, ray, angle, • Line, line segment, • Point, line, line line segment endpoint segment, endpoint •Properties of parallel, perpen- • Parallel, perpendicular dicular, and intersecting lines • Angles – acute, right, obtuse •Sum of the measures of the [Students in early • Circles – diameter, interior angles of a triangle is elementary grades radius, center 180° sometimes confuse space- 4. Recognize, describe, extend 5. Recognize, describe, 5. Recognize, describe, filling patterns (discussed and create designs and patterns extend, and create space- extend, and create space- here) with sequential with geometric objects of filling patterns. filling patterns. patterns discussed in different shapes and colors. Preschool Mathematics Expectations 3.5 and 3.6 and in Standard 4.3.] Adopted January 9, 2008 197 Adopted January 9, 2008 4.2 GEOMETRY AND MEASUREMENT Descriptive Statement: Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve describing the shapes we see all around us in art, nature, and the things we make. Spatial sense, geometric modeling, and measurement can help us to describe and interpret our physical environment and to solve problems. 4.2.6 A. Geometric Properties 4.2.7 A. Geometric Properties 4.2.8 A. Geometric Properties 4.2.12 A. Geometric Properties Grade 6 Grade 7 Grade 8 Grade 12 Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the end of Grade 6, students will: end of Grade 7, students will: end of Grade 8, students will: end of Grade 12, students will: 6. Identify, describe, and draw the faces or shadows (projections) of three-dimensional geometric 7. Create two-dimensional objects from different perspectives. representations (e.g., nets 7. Identify a three-dimensional shape or projective views) for the 2. Draw perspective views of with given projections (top, front surfaces of three- 3D objects on isometric dot and side views). dimensional objects. paper, given 2D 8. Identify a three-dimensional shape representations (e.g., nets or with a given net (i.e., a flat pattern projective views). that folds into a 3D shape). 2. Identify, describe, compare, 1. Understand and apply 3. Understand and apply 1. Use geometric models to and classify polygons and properties of polygons. properties of polygons. represent real-world circles. • Quadrilaterals, including • Quadrilaterals, including situations and objects and to • Triangles by angles and sides squares, rectangles, squares, rectangles, parallelo- solve problems using those • Quadrilaterals, including parallelograms, trapezoids, grams, trapezoids, rhombi models (e.g., use Pythagorean squares, rectangles, parallelo- rhombi • Regular polygons Theorem to decide whether grams, trapezoids, rhombi • Sum of measures of interior • Regular polygons an object can fit through a • Polygons by number of sides angles of a polygon doorway). • Equilateral, equiangular, regular • Which polygons can be used • All points equidistant from a alone to generate a tessellation given point form a circle and why 5. Compare properties of cylinders, prisms, cones, pyramids, and spheres. 2. Understand and apply the Pythagorean theorem. 3. Identify similar figures. 2. Understand and apply the 4. Understand and apply the concept of similarity. concept of similarity. • Using proportions to find • Using proportions to find missing measures missing measures • Scale drawings • Scale drawings • Models of 3D objects • Models of 3D objects 4. Understand and apply the 3. Apply the properties of geometric shapes. concepts of congruence and • Parallel lines – transversal, alternate symmetry (line and rotational). interior angles, corresponding angles • Triangles 1. Understand and apply concepts 1. Understand and apply concepts a. Conditions for congruence involving lines and angles. involving lines, angles, and planes. b. Segment joining midpoints of • Notation for line, ray, • Complementary and two sides is parallel to and half angle, line segment supplementary angles the length of the third side • Properties of parallel, perpen- • Vertical angles c. Triangle Inequality dicular, and intersecting lines • Bisectors and perpendicular bisectors • Minimal conditions for a shape to • Parallel, perpendicular, and be a special quadrilateral • Sum of the measures of the intersecting planes • Circles – arcs, central and inscribed interior angles of a triangle • Intersection of plane with cube, angles, chords, tangents is 180° cylinder, cone, and sphere • Self-similarity 6. Perform basic geometric 5. Perform basic geometric constructions using a variety of constructions using a variety of methods (e.g., straightedge and methods (e.g., straightedge and compass, compass, patty/tracing paper, or patty/tracing paper, or technology). technology). • Perpendicular bisector of a line segment • Congruent angles or line segments • Bisector of an angle • Midpoint of a line segment • Perpendicular or parallel lines 3. Use logic and reasoning to 5. Use logic and reasoning to 4. Use reasoning and some form of proof to make and support conjectures make and support conjectures verify or refute conjectures and theorems. about geometric objects. about geometric objects. • Verification or refutation of proposed proofs • Simple proofs involving congruent triangles • Counterexamples to incorrect conjectures Adopted January 9, 2008 198 Adopted January 9, 2008 Transforming Shapes. Analyzing how various transformations affect geometric objects allows students to enhance their spatial sense. This includes combining shapes to form new ones and decomposing complex shapes into simpler ones. It includes the standard geometric transformations of translation (slide), reflection (flip), rotation (turn), and dilation (scaling). It also includes using tessellations and fractals to create geometric patterns. Preschool Learning 4.2.2 B. Transforming 4.2.3 B. Transforming 4.2.4 B. Transforming 4.2.5 B. Transforming Expectations Shapes Grade 2 Shapes Grade 3 Shapes Grade 4 Shapes Grade 5 [Identifying patterns is included 1. Use simple shapes to 1. Use simple shapes to in Preschool Mathematics make designs, patterns, cover an area Expectation 3.5 below] and pictures. (tessellations). 2. Combine and subdivide simple shapes to make other shapes. 1. Describe and use 2. Describe and use 1. Use a translation, a geometric geometric reflection, or a rotation transformations transformations to map one figure onto (slide, flip, turn). (slide, flip, turn). another congruent figure. 3.5 Identifies patterns 2. Investigate the 3. Investigate the 2. Recognize, identify, and in the environment occurrence of geometry occurrence of geometry describe geometric (e.g., "Look at the rug. in nature and art. in nature and art. relationships and It has a circle, then a properties as they exist number, then a in nature, art, and other letter..."). real-world settings. Coordinate Geometry. Coordinate geometry provides an important connection between geometry and algebra. It facilitates the visualization of algebraic relationships, as well as an analytical understanding of geometry. Preschool Learning 4.2.2 C. Coordinate 4.2.3 C. Coordinate 4.2.4 C. Coordinate 4.2.5 C. Coordinate Expectations Geometry Grade 2 Geometry Grade 3 Geometry Grade 4 Geometry Grade 5 1. Locate and name points 1. Locate and name points 1. Create geometric shapes in the first quadrant on a in the first quadrant on a with specified properties coordinate grid. coordinate grid. in the first quadrant on a coordinate grid. [Vocabulary to describe distances is 1. Give and follow directions 2. Use coordinates to give or included in Preschool Mathematics for getting from one point follow directions from one point Expectation 2.3 below] to another on a map or grid. to another on a map or grid. 2.4 Uses vocabulary to describe directional concept (e.g., "Watch me climb up the ladder and slide down."). Adopted January 9, 2008 199 Adopted January 9, 2008 4.2 GEOMETRY AND MEASUREMENT 4.2.6 B. Transforming Shapes 4.2.7 B. Transforming Shapes 4.2.8 B. Transforming Shapes 4.2.12 B. Transforming Shapes Grade 6 Grade 7 Grade 8 Grade 12 [Determining which polygons can be 3. Determine whether two or used alone to generate a tessellation more given shapes can be is included in indicator 4.2.8 A 3.] used to generate a tessellation. [Finding the area of geometric figures made by combining other figures is included in indicators 4.2.7 E 1 and 4.2.8 E 1.] 1. Use a translation, a reflection, 2. Understand and apply 1. Understand and apply 1. Determine, describe, and draw the or a rotation to map one transformations. transformations. effect of a transformation, or a sequence figure onto another congruent • Finding the image, given the • Finding the image, given the of transformations, on a geometric or figure. pre-image, and vice-versa pre-image, and vice-versa algebraic [object] representation, and, • Sequence of transformations • Sequence of transformations conversely, determine whether and how needed to map one figure onto needed to map one figure onto one [object]representation can be another another transformed to another by a • Reflections, rotations, and • Reflections, rotations, and transformation or a sequence of translations result in images translations result in images transformations. congruent to the pre-image congruent to the pre-image 2. Recognize three-dimensional • Dilations (stretching/shrinking) • Dilations (stretching/shrinking) figures obtained through trans- result in images similar to the result in images similar to the formations of two-dimensional pre-image pre-image figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization. 2. Recognize, identify, and 2. Use iterative procedures to 4. Generate and analyze describe geometric generate geometric patterns. iterative geometric patterns. relationships and properties • Fractals (e.g., the Koch • Fractals (e.g., Sierpinski’s as they exist in nature, art, Snowflake) Triangle) and other real-world settings. • Self-similarity • Patterns in areas and • Construction of initial stages perimeters of self-similar • Patterns in successive stages figures (e.g., number of triangles in each stage of Sierpinski’s • Outcome of extending Triangle) iterative process indefinitely 4.2.6 C. Coordinate Geometry 4.2.7 C. Coordinate Geometry 4.2.8 C. Coordinate Geometry 4.2.12 C. Coordinate Geometry Grade 6 Grade 7 Grade 8 Grade 12 1. Create geometric shapes with 1. Use coordinates in four 1. Use coordinates in four 1. Use coordinate geometry to specified properties in the first quadrants to represent quadrants to represent represent and verify properties quadrant on a coordinate grid. geometric concepts. geometric concepts. of lines and line segments. • Distance between two points • Midpoint and slope of a line segment • Finding the intersection of two lines [Graphing functions on the [Developing an informal • Lines with the same slope are parallel coordinate plane is included notion of slope is included • Lines that are perpendicular have in indicator 4.3.7 B 1.] in indicator 4.3.8 B 1.] slopes whose product is –1 2. Use a coordinate grid to model 2. Use a coordinate grid to model and quantify transformations and quantify transformations (e.g., translate right 4 units). (e.g., translate right 4 units). 2. Show position and represent motion in the coordinate plane using vectors. • Addition and subtraction of vectors 3. Find an equation of a circle given its center and radius and, given an equation of a circle in standard form, find its center and radius. Adopted January 9, 2008 200 Adopted January 9, 2008 Units of Measurement. Measurement helps describe our world using numbers. An understanding of how we attach numbers to real-world phenomena, familiarity with common measurement units (e.g., inches, liters, and miles per hour), and a practical knowledge of measurement tools and techniques are critical for students' understanding of the world around them. Preschool Learning 4.2.2 D. Units of 4.2.3 D. Units of 4.2.4 D. Units of 4.2.5 D. Units of Expectations Measurement Grade 2 Measurement Grade 3 Measurement Grade 4 Measurement Grade 5 3.4 Seriates objects 1. Directly compare and 1. Understand that everyday 1. Understand that everyday [Relate to science indicator according to various order objects according to objects have a variety of objects have a variety of 5.3.4 A 1, determining the properties including size, measurable attributes. attributes, each of which attributes, each of which reasonableness of estimates, number, length, • Attributes – length, can be measured in many can be measured in many measurements, and heaviness, texture (rough weight, capacity, time, ways. ways. computations when doing to smooth) or loudness. temperature science.] 2. Recognize the need for a uniform unit of measure. 2.2 Uses standard and 3. Select and use appropriate 2. Select and use appropriate 2. Select and use appropriate 1. Select and use nonstandard measurement standard and non-standard standard units of measure standard units of measure appropriate units to units (e.g., measuring units of measure and and measurement tools to and measurement tools to measure angles and area. body length with unifix standard measurement tools solve real-life problems. solve real-life problems cubes, using a tape to solve real-life problems. measure to gauge height • Length – inch, foot, • Length – fractions of an • Length – fractions of an of block construction, yard, centimeter, meter inch (1/4, 1/2), mile, inch (1/8, 1/4, 1/2), mile, counting the number of decimeter, kilometer decimeter, kilometer cups it takes to fill a • Area – square inch, • Area – square inch, bucket with water). square centimeter square centimeter • Volume – cubic inch, cubic centimeter • Weight – pound, gram, • Weight – ounce • Weight – ounce kilogram • Capacity – pint, quart, • Capacity – fluid ounce, • Capacity – fluid ounce, liter cup, gallon, milliliter cup, gallon, milliliter • Time – second, minute, 5. Solve problems hour, day, week, month, involving elapsed time. year • Temperature – degrees Celsius, degrees Fahrenheit 2. Convert measurement units within a system (e.g., 3 feet = __ inches). 3. Develop and use personal 3. Know approximate referents to approximate equivalents between the standard units of measure standard and metric systems (e.g., a common paper (e.g., one kilometer is clip is about an inch long). approximately 6/10 of a mile). [Using estimation as a 4. Estimate measures. 3. Incorporate estimation 4. Incorporate estimation in method for approximating in measurement measurement activities (e.g., an appropriate amount is activities (e.g., estimate estimate before measuring). included in Preschool before measuring). [Relate to science indicator Mathematics Expectation 5.3.4 B 1 Select appropriate 1.7 above] measuring instruments based on the degree of precision required.] 2.3 Uses vocabulary to 4. Use measurements and describe distances estimates to describe (e.g., "It was a really and compare long walk to the phenomena. playground."). Adopted January 9, 2008 201 Adopted January 9, 2008 4.2 GEOMETRY AND MEASUREMENT 4.2.6 D. Units of Measurement 4.2.7 D. Units of Measurement 4.2.8 D. Units of Measurement 4.2.12 D. Units of Measurement Grade 6 Grade 7 Grade 8 Grade 12 1. Select and use appropriate units to measure angles, area, surface area, and volume. 2. Use a scale to find a distance on a map or a length on a scale drawing. 1. Solve problems requiring 1. Solve problems requiring calculations that involve calculations that involve 3. Convert measurement units different units of measurement different units of measurement within a system (e.g., within a measurement system within a measurement system 3 feet = ___ inches). (e.g., 4’3” plus 7’10” equals (e.g., 4’3” plus 7’10” equals 12’1”). 12’1”). 4. Know approximate equivalents 2. Use approximate equivalents between the standard and between standard and metric metric systems (e.g., one systems to estimate kilometer is approximately measurements (e.g., 5 6/10 of a mile). kilometers is about 3 miles). 3. Recognize that all 5. Recognize that all 1. Understand and use the measurements of continuous measurements of continuous concept of significant digits. quantities are approximations. quantities are approximations. 3. Recognize that the degree 2. Choose appropriate tools and of precision needed in techniques to achieve the calculations depends on how specified degree of precision the results will be used and and error needed in a the instruments used to situation. generate the measurements. • Degree of accuracy of a given 5. Use measurements and 2. Select and use appropriate units 4. Select and use appropriate measurement tool estimates to describe and and tools to measure quantities units and tools to measure • Finding the interval in which a compare phenomena. to the degree of precision quantities to the degree of computed measure (e.g., area needed in a particular problem- precision needed in a or volume) lies, given the solving situation. particular problem-solving degree of precision of linear situation. measurements 6. Solve problems that involve compound measurement units, such as speed (miles per hour), air pressure (pounds per square inch), and population density (persons per square mile). Adopted January 9, 2008 202 Adopted January 9, 2008 Measuring Geometric Objects. This area focuses on applying the knowledge and understandings of units of measurement in order to actually perform measurement. While students will eventually apply formulas, it is important that they develop and apply strategies that derive from their understanding of the attributes. In addition to measuring objects directly, students apply indirect measurement skills, using, for example, similar triangles and trigonometry. Preschool Learning 4.2.2 E. Measuring 4.2.3 E. Measuring 4.2.4 E. Measuring 4.2.5 E. Measuring Expectations Geometric Objects Geometric Objects Geometric Objects Geometric Objects Grade 2 Grade 3 Grade 4 Grade 5 [Use of nonstandard measure- 2. Directly measure the area 1. Determine the area of 1. Determine the area of ment units is included in of simple two-dimensional simple two-dimensional simple two-dimensional Preschool Mathematics shapes by covering them shapes on a square grid. shapes on a square grid. Expectation 2.2 above] with squares. 1. Use a protractor to [Relate to Science Indicator measure angles. 5.3.4 B 2 Use a variety of measuring instruments and record measured quantities using the appropriate units.] 1. Directly measure the 2. Determine the perimeter 2. Distinguish between 2. Develop and apply perimeter of simple of simple shapes by perimeter and area and strategies and formulas two-dimensional shapes. measuring all of the use each appropriately for finding perimeter sides. in problem-solving and area. situations. • Square • Rectangle 3. Recognize that rectangles with the same perimeter do not necessarily have the same area and vice versa. [Comparing numbers in 3. Measure and compare 3. Measure and compare context (e.g., using words the volume of the volume of such as more and less) is three-dimensional three-dimensional included in Preschool objects using materials objects using materials Mathematics Expectation 1.6 such as rice or cubes. such as rice or cubes. above] 4. Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one’s foot). Adopted January 9, 2008 203 Adopted January 9, 2008 4.2 GEOMETRY AND MEASUREMENT 4.2.6 E. 4.2.7 E. 4.2.8 E. 4.2.12 E. Measuring Geometric Objects Measuring Geometric Objects Measuring Geometric Objects Measuring Geometric Objects Grade 6 Grade 7 Grade 8 Grade 12 [Finding area is included in indicators 4.2.6 E 2 and 4.2.7 E 1 below.] 1. Use a protractor to 1. Use techniques of indirect measure angles. measurement to represent and solve problems. • Similar triangles • Pythagorean theorem • Right triangle trigonometry (sine, cosine, tangent) • Special right triangles 2. Develop and apply strategies 1. Develop and apply strategies 1. Develop and apply strategies 2. Use a variety of strategies to and formulas for finding for finding perimeter and area. for finding perimeter and determine perimeter and area perimeter and area. • Geometric figures made by area. of plane figures and surface • Triangle, square, rectangle, combining triangles, • Geometric figures made by area and volume of 3D figures. parallelogram, and trapezoid rectangles and circles or combining triangles, • Approximation of area using • Circumference and area of a parts of circles rectangles and circles or grids of different sizes circle • Estimation of area using parts of circles • Finding which shape has grids of various sizes • Estimation of area using minimal (or maximal) area, grids of various sizes perimeter, volume, or surface 4. Recognize that shapes with area under given conditions • Impact of a dilation on the the same perimeter do not using graphing calculators, perimeter and area of a necessarily have the same dynamic geometric software, 2-dimensional figure area and vice versa. and/or spreadsheets • Estimation of area, perimeter, volume, and surface area 2. Recognize that the volume of 2. Recognize that the volume of [Relate to indicator 4.2.12 B 2, a pyramid or cone is one-third a pyramid or cone is one-third recognizing three-dimensional of the volume of the prism or of the volume of the prism or figures obtained through trans- cylinder with the same base cylinder with the same base formations of two-dimensional and height (e.g., use rice to and height (e.g., use rice to figures (e.g., cone as rotating an compare volumes of figures compare volumes of figures isosceles triangle about an with same base and height). with same base and height). altitude)] 3. Develop and apply strategies 3. Develop and apply strategies and formulas for finding the and formulas for finding the surface area and volume of surface area and volume of a rectangular prisms and three-dimensional figure. cylinders. • Volume - prism, cone, pyramid [Finding surface area and • Surface area - prism (triangular volume of 3D figures is or rectangular base), pyramid included in indicator (triangular or rectangular 4.2.12 E 2 above.] base) • Impact of a dilation on the surface area and volume of a three-dimensional figure 4. Use formulas to find the volume and surface area of a sphere. 5. Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one’s foot). Students of all ages should realize that geometry and measurement are all around them. Through study of these areas and their applications, they should come to better understand and appreciate the role of mathematics in their lives. Adopted January 9, 2008 204 Adopted January 9, 2008 STANDARD 4.3 (PATTERNS AND ALGEBRA) ALL STUDENTS WILL REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES AND SOLVE PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES. Patterns. Algebra provides the language through which we communicate the patterns in mathematics. From the earliest age, students should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and, by doing so, to make mathematical discoveries. They should have opportunities to analyze, extend, and create a variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena. Preschool Learning 4.3.2 A. Patterns 4.3.3 A. Patterns 4.3.4 A. Patterns 4.3.5 A. Patterns Expectations Grade 2 Grade 3 Grade 4 Grade 5 EXPECTATION 3: By the end of Grade 2, Building upon knowledge Building upon knowledge Building upon knowledge Children understand students will: and skills gained in and skills gained in and skills gained in patterns, relationships preceding grades, by the end preceding grades, by the end preceding grades, by the end of Grade 3, students will: of Grade 4, students will: of Grade 5, students will: and classification. 3.5 Identifies patterns in 1. Recognize, describe, 1. Recognize, describe, 1. Recognize, describe, 1. Recognize, describe, the environment extend, and create extend, and create extend, and create extend, and create (e.g., "Look at the rug. It patterns. patterns. patterns. patterns involving has a circle, then a number, whole numbers. then a letter..."). 3.6 Represents patterns in • Using concrete a variety of ways materials (e.g., stringing beads (manipulatives), red/green/red/green/red/green, pictures, rhythms, & arranging buttons whole numbers big/bigger/biggest, or singing songs that follow a simple pattern). • Descriptions using • Descriptions using • Descriptions using Descriptions using words and symbols words and number words, number tables, verbal rules, (e.g., “add two” or “+ sentences/expressions sentences/expressions, simple equations, and 2”) graphs, tables, variables graphs (e.g., shape, blank, or letter) • Repeating patterns • Sequences that stop or that continue infinitely • Whole number Whole number patterns • Whole number patterns patterns that grow or that grow or shrink as that grow or shrink as shrink as a result of a result of repeatedly a result of repeatedly repeatedly adding or adding, subtracting, adding, subtracting, subtracting a fixed multiplying by, or multiplying by, or number (e.g., skip dividing by a fixed dividing by a fixed counting forward or number number backward) (e.g., 5, 8, 11, . . . or (e.g., 5, 8, 11, . . . or 800, 400, 200, . . .) 800, 400, 200, . . .) • Sequences can often [Use of calculators to be extended in more explore patterns is included in indicator than one way (e.g., 4.5 F 4.] the next term after 1, 2, 4, . . . could be 8, or 7, or … ) Adopted January 9, 2008 205 Adopted January 9, 2008 4.3 PATTERNS AND ALGEBRA Descriptive Statement: Algebra is a symbolic language used to express mathematical relationships. Students need to understand how quantities are related to one another, and how algebra can be used to concisely express and analyze those relationships. Modern technology provides tools for supplementing the traditional focus on algebraic procedures, such as solving equations, with a more visual perspective, with graphs of equations displayed on a screen. Students can then focus on understanding the relationship between the equation and the graph, and on what the graph represents in a real-life situation. 4.3.6 A. Patterns 4.3.7 A. Patterns 4.3.8 A. Patterns 4.3.12 A. Patterns Grade 6 Grade 7 Grade 8 Grade 12 Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the end of Grade 6, students will: end of Grade 7, students will: end of Grade 8, students will: end of Grade 12, students will: 1. Recognize, describe, extend, 1. Recognize, describe, extend, 1. Recognize, describe, extend, 1. Use models and algebraic and create patterns involving and create patterns involving and create patterns involving formulas to represent and whole numbers and rational whole numbers, rational whole numbers, rational analyze sequences and series. numbers. numbers, and integers. numbers, and integers. • Descriptions using tables, • Descriptions using tables, • Descriptions using tables, • Explicit formulas for nth verbal rules, simple verbal and symbolic rules, verbal and symbolic rules, terms equations, and graphs graphs, simple equations or graphs, simple equations or expressions expressions • Formal iterative formulas • Finite and infinite sequences • Finite and infinite sequences (e.g., NEXT = NOW * 3) • Recursive patterns, • Arithmetic sequences • Sums of finite arithmetic including Pascal’s Triangle (i.e., sequences generated by series (where each entry is the sum repeated addition of a fixed of the entries above it) and number, positive or negative) the Fibonacci Sequence: • Geometric sequences • Sums of finite and infinite 1, 1, 2, 3, 5, 8, . . . (where (i.e., sequences generated by geometric series NEXT = NOW + PREVIOUS) repeated multiplication by a fixed positive ratio, greater than 1 or less than 1) • Generating sequences by • Generating sequences by using calculators to using calculators to repeatedly apply a formula repeatedly apply a formula 2. Develop an informal notion of limit. 3. Use inductive reasoning to form generalizations. Adopted January 9, 2008 206 Adopted January 9, 2008 Functions and Relationships. The function concept is one of the most fundamental unifying ideas of modern mathematics. Students begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships. 4.3.2 B. 4.3.3 B. 4.3.4 B. 4.3.5 B. Functions and Relationships Functions and Relationships Functions and Relationships Functions and Relationships Grade 2 Grade 3 Grade 4 Grade 5 1. Use concrete and pictorial 1. Use concrete and pictorial 1. Use concrete and pictorial 2. Graph points satisfying a models of function models to explore the basic models to explore the basic function from T-charts, machines to explore the concept of a function. concept of a function. from verbal rules, and from basic concept of a function. simple equations. • Input/output tables, T-charts • Input/output tables, T- charts No Associated Preschool Learning Expectations • Combining two function 1. Describe arithmetic machines operations as functions, • Reversing a function including combining machine operations and reversing them. [Transformations [Translations and are introduced in reflections are introduced indicator 4.2.3 B 1 in indicator 4.2.5 B 1 above] above] Adopted January 9, 2008 207 Adopted January 9, 2008 4.3 PATTERNS AND ALGEBRA 4.3.6 B. 4.3.7 B. 4.3.8 B. 4.3.12 B. Functions and Relationships Functions and Relationships Functions and Relationships Functions and Relationships Grade 6 Grade 7 Grade 8 Grade 12 1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs. 1. Describe the general 1. Graph functions, and 1. Graph functions, and 2. Analyze and explain the general behavior of functions given understand and describe understand and describe properties and behavior of functions by formulas or verbal rules their general behavior. their general behavior. [of one variable] or relations, using (e.g., graph to determine [appropriate] algebraic and whether increasing or • Equations involving two • Equations involving two graphing [technologies] techniques. decreasing, linear or not). variables variables • Rates of change (informal • Slope of a line [or curve] notion of slope) • Domain and range • Intercepts • Continuity • Maximum/minimum • Estimating roots of equations • [Intersecting points as] Solutions of systems of equations • Solutions of systems of linear inequalities using graphing techniques • Rates of change 3. Understand and perform transformations on commonly-used functions. • Translations, reflections, dilations • Effects on linear and quadratic graphs of parameter changes in equations • Using graphing calculators or computers for more complex functions 2. Recognize and describe the 4. Understand and compare the difference between linear properties of classes of functions, and exponential growth, including exponential, polynomial, using tables, graphs, and rational, and trigonometric equations. functions. • Linear vs. non-linear • Symmetry • Increasing/decreasing on an interval Adopted January 9, 2008 208 Adopted January 9, 2008 Modeling. Algebra is used to model real situations and answer questions about them. This use of algebra requires the ability to represent data in tables, pictures, graphs, equations or inequalities, and rules. Modeling ranges from writing simple number sentences to help solve story problems in the primary grades to using functions to describe the relationship between two variables, such as the height of a pitched ball over time. Modeling also includes some of the conceptual building blocks of calculus, such as how quantities change over time and what happens in the long run (limits). Preschool Learning 4.3.2 C. Modeling 4.3.3 C. Modeling 4.3.4 C. Modeling 4.3.5 C. Modeling Expectations Grade 2 Grade 3 Grade 4 Grade 5 4.2 Describes the sequence of 1. Recognize and describe 1. Recognize and describe 1. Recognize and describe 2. Draw freehand sketches the daily routine and changes over time (e.g., change in quantities. change in quantities. of graphs that model real demonstrates understanding temperature, height). • Graphs representing • Graphs representing phenomena and use such of basic temporal relations change over time (e.g., change over time (e.g., graphs to predict and (e.g., "We will go outside temperature, height) temperature, height) interpret events. after snack time."). • How change in one • Changes over time physical quantity can • Rates of change (e.g., [Understanding that living things change as they grow produce a corresponding when is plant growing is included in Preschool change in another (e.g., slowly/rapidly, when is Science Expectation 3.3] pitch of a sound depends temperature dropping on the rate of vibration) most rapidly/slowly) 2. Construct and solve 2. Construct and solve 2. Construct and solve 1. Use number sentences to simple open sentences simple open sentences simple open sentences model situations. involving addition or involving addition or involving any one • Using variables to subtraction. subtraction operation represent unknown • Result unknown (e.g., 3 + 6 = __, (e.g., 3 x 6 = __, quantities (e.g., 6 – 2 = __ or n = 15 – 3, n = 15 ÷ 3, • Using concrete materials, n = 3 + 5) 3 + __ = 3, 3 x __ = 0, tables, graphs, verbal • Part unknown 16 – c = 7). 16 – c = 7). rules, algebraic (e.g., 3 + = 8) expressions/equations Adopted January 9, 2008 209 Adopted January 9, 2008 4.3 PATTERNS AND ALGEBRA 4.3.6 C. Modeling 4.3.7 C. Modeling 4.3.8 C. Modeling 4.3.12 C. Modeling Grade 6 Grade 7 Grade 8 Grade 12 2. Draw freehand sketches of 1. Analyze functional 1. Analyze functional 2. Analyze and describe how a graphs that model real pheno- relationships to explain how a relationships to explain how a change in an independent mena and use such graphs to change in one quantity can change in one quantity can variable leads to change in a predict and interpret events. result in a change in another, result in a change in another, dependent one. • Changes over time using pictures, graphs, charts, using pictures, graphs, charts, • Relations between quantities and equations. and equations. • Rates of change (e.g., when 3. Convert recursive formulas to is plant growing linear or exponential slowly/rapidly, when is functions (e.g., Tower of temperature dropping most Hanoi and doubling). rapidly/slowly) 1. Use patterns, relations, and 2. Use patterns, relations, 2. Use patterns, relations, 1. Use functions to model real- linear functions to model symbolic algebra, and linear symbolic algebra, and linear world phenomena and solve situations. functions to model situations. functions to model situations. problems that involve varying • Using variables to represent • Using manipulatives, tables, • Using concrete materials quantities. unknown quantities graphs, verbal rules, (manipulatives), tables, • Linear, quadratic, exponential, • Using concrete materials, algebraic expressions/ graphs, verbal rules, periodic (sine and cosine), and tables, graphs, verbal rules, equations/inequalities algebraic expressions/ step functions (e.g., price of algebraic expressions/ equations/inequalities mailing a first-class letter over equations/inequalities • Growth situations, such as • Growth situations, such as the past 200 years) population growth and population growth and • Direct and inverse variation compound interest, using compound interest, using • Absolute value recursive (e.g., NOW- recursive (e.g., NOW- • Expressions, equations and NEXT) formulas (cf. NEXT) formulas (cf. science inequalities science standards and social standards and social studies • Same function can model studies standards) standards) variety of phenomena • Growth/decay and change in the natural world • Applications in mathematics, biology, and economics (including compound interest) Adopted January 9, 2008 210 Adopted January 9, 2008 Procedures. Techniques for manipulating algebraic expressions – procedures – remain important, especially for students who may continue their study of mathematics in a calculus program. Utilization of algebraic procedures includes understanding and applying properties of numbers and operations, using symbols and variables appropriately, working with expressions, equations, and inequalities, and solving equations and inequalities. Preschool Learning 4.3.2 D. Procedures 4.3.3 D. Procedures 4.3.4 D. Procedures 4.3.5 D. Procedures Expectations Grade 2 Grade 3 Grade 4 Grade 5 [Use of a number line [Use of a number line [Use of a number line to construct meanings to construct meanings to construct meanings for numbers at this for numbers at this for numbers at this grade level is included grade level is included grade level is included in indicator 4.1.3 A 1.] in indicators 4.1.4 A 1 in indicator 4.1.5 A 1.] and 4.1.4 A 7.] 1. Solve simple linear equations with manipulatives and informally • Whole-number coefficients only, answers also whole numbers • Variables on one side of equation [Comparing numbers in 2. Understand and use the 2. Understand and use the different contexts (e.g., using concepts of equals, less concepts of equals, less words such as more and less) than, and greater than to than, and greater than in is included in Preschool describe relations simple number Mathematics Expectation 1.6 between numbers. sentences. above] • Symbols ( = , < , > ) • Symbols ( = , < , > ) 1. Understand and apply 1. Understand and apply 1. Understand, name, and (but don’t name) the the properties of apply the properties of following properties of operations and numbers. operations and numbers. addition: • Commutative • Commutative • Commutative (e.g., 5 + 3 = 3 + 5) (e.g., 3 x 7 = 7 x 3) (e.g., 3 x 7 = 7 x 3) • Zero as the identity • Identity element for • Identity element for element multiplication is 1 multiplication is 1 (e.g., 7 + 0 = 7) (e.g., 1 x 8 = 8) (e.g., 1 x 8 = 8) • Associative (e.g., • Associative (e.g., 7 + 3 + 2 can be found 2 x 4 x 25 can be by first adding either found by first 7 + 3 or 3 + 2) multiplying either 2 x 4 or 4 x 25) • Division by zero is undefined • Any number multiplied • Any number by zero is zero multiplied by zero is zero Adopted January 9, 2008 211 Adopted January 9, 2008 4.3 PATTERNS AND ALGEBRA 4.3.6 D. Procedures 4.3.7 D. Procedures 4.3.8 D. Procedures 4.3.12 D. Procedures Grade 6 Grade 7 Grade 8 Grade 12 1. Use graphing techniques on a 1. Use graphing techniques on a [Use of a number line to number line. number line. construct meanings for • Absolute value • Absolute value numbers at this grade level • Arithmetic operations • Arithmetic operations is included in indicator represented by vectors represented by vectors 4.1.6 A 1.] (arrows) (arrows) (e.g., “-3 + 6” is “left 3, (e.g., “-3 + 6” is “left 3, right 6”) right 6”) 1. Solve simple linear equations 2. Solve simple linear equations 2. Solve simple linear equations 2. Select and use appropriate with manipulatives and informally and graphically. informally, graphically, and methods to solve equations and informally. • Multi-step, integer using formal algebraic inequalities. • Whole-number coefficients coefficients only (although methods. • Linear equations and only, answers also whole answers may not be • Multi-step, integer inequalities – algebraically numbers integers) coefficients only (although • Quadratic equations – factoring • Variables on one or both • Using paper-and-pencil, answers may not be integers) (including trinomials when sides of equation calculators, graphing • Simple literal equations the coefficient of x2 is 1) and calculators, spreadsheets, (e.g., A = lw) using the quadratic formula and other technology • Literal equations • Using paper-and-pencil, • All types of equations and calculators, graphing calculators, spreadsheets, inequalities using graphing, computer, and graphing and other technology calculator techniques 4. Extend understanding and use 3. Solve simple linear of inequality. inequalities. [Use of concrete representations • Symbols ( ≥ , ≠ , ≤ ) (e.g., algebra tiles) is included in indicator 4.5 E 1.] 2. Understand and apply the 4. Understand and apply the 5. Understand and apply the properties of operations and properties of operations, properties of operations, numbers. numbers, equations, and numbers, equations, and • Distributive property inequalities. inequalities. • The product of a number • Additive inverse • Additive inverse and its reciprocal is 1 • Multiplicative inverse • Multiplicative inverse • Addition and multiplication properties of equality • Addition and multiplication properties of inequalities 3. Evaluate numerical 3. Create, evaluate, and simplify 4. Create, evaluate, and simplify 1. Evaluate and simplify expressions. algebraic expressions algebraic expressions expressions. involving variables. involving variables. • Add and subtract • Order of operations, including • Order of operations, including polynomials [The distributive property appropriate use of parentheses appropriate use of parentheses • Multiply a polynomial by a appears in 4.3.6 D 2 above.] • Substitution of a number for a • Distributive property monomial or binomial variable • Substitution of a number for a • Divide a polynomial by a variable monomial • Translation of a verbal phrase • Perform simple operations or sentence into an algebraic with rational expressions expression, equation, or • Evaluate polynomial and inequality, and vice versa rational expressions 3. Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. Algebra is a gatekeeper for the future study of mathematics, science, the social sciences, business, and a host of other areas. In the past, algebra has served as a filter, screening people out of these opportunities. For New Jersey to be part of the global society, it is important that algebra play a major role in a mathematics program that opens the gates for all students. Adopted January 9, 2008 212 Adopted January 9, 2008 STANDARD 4.4 (DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS) ALL STUDENTS WILL DEVELOP AN UNDERSTANDING OF THE CONCEPTS AND TECHNIQUES OF DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS, AND WILL USE THEM TO MODEL SITUATIONS, SOLVE PROBLEMS, AND ANALYZE AND DRAW APPROPRIATE INFERENCES FROM DATA. Data Analysis (or Statistics). In today’s information-based world, students need to be able to read, understand, and interpret data in order to make informed decisions. In the early grades, students should be involved in collecting and organizing data, and in presenting it using tables, charts, and graphs. As they progress, they should gather data using sampling, and should increasingly be expected to analyze and make inferences from data, as well as to analyze data and inferences made by others. Preschool Learning 4.4.2 A. Data Analysis 4.4.3 A. Data Analysis 4.4.4 A. Data Analysis 4.4.5 A. Data Analysis Expectations Grade 2 Grade 3 Grade 4 Grade 5 EXPECTATION 4: By the end of Grade 2, Building upon knowledge and Building upon knowledge and Building upon knowledge and Children develop students will: skills gained in preceding skills gained in preceding skills gained in preceding knowledge of sequence grades, by the end of Grade 3, grades, by the end of Grade 4, grades, by the end of Grade 5, and temporal awareness. students will: students will: students will: 1. Collect, generate, record, 1. Collect, generate, 1. Collect, generate, 1. Collect, generate, [Classifying objects by sorting and organize data in organize, and display data organize, and display data organize, and display them into subgroups by one or response to questions, in response to questions, in response to questions, data. more attributes is included in claims, or curiosity. claims, or curiosity. claims, or curiosity. Preschool Mathematics Expectation 3.2 below] • Data collected from students’ • Data collected from the • Data collected from the • Data generated from everyday experiences classroom environment school environment surveys • Data generated from chance devices, such as spinners and dice 4.3 Arranges pictures of events in 2. Read, interpret, construct, 2. Read, interpret, construct, 2. Read, interpret, construct, 2. Read, interpret, select, temporal order (e.g., first, a photo and analyze displays of analyze, generate analyze, generate construct, analyze, of the child eating breakfast; data. questions about, and draw questions about, and draw generate questions about, second, a photo of the child inferences from displays inferences from displays and draw inferences from getting on the bus; third, a photo of data. of data. displays of data. of the child in the classroom). • Pictures, tally chart, pictograph, • Pictograph, bar graph, • Pictograph, bar graph, • Bar graph, line graph, bar graph, Venn diagram table line plot, line graph, table circle graph, table [Seriating objects according • Smallest to largest, • Average (mean), • Range, to various properties most frequent (mode) most frequent (mode), median, and including size, number, middle term (median) mean length, heaviness, texture [Interpreting information in (rough to smooth) or loudness graphs, charts, and is included in Preschool diagrams is included in Mathematics Expectation 3.4 language arts literacy below] indicator 3.1.3 G 3] 3. Respond to questions about data and generate their own questions and hypotheses. Adopted January 9, 2008 213 Adopted January 9, 2008 4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS Descriptive Statement: Data analysis, probability, and discrete mathematics are important interrelated areas of applied mathematics. Each provides students with powerful mathematical perspectives on everyday phenomena and with important examples of how mathematics is used in the modern world. Two important areas of discrete mathematics are addressed in this standard; a third area, iteration and recursion, is addressed in Standard 4.3 (Patterns and Algebra). 4.4.6 A. Data Analysis 4.4.7 A. Data Analysis 4.4.8 A. Data Analysis 4.4.12 A. Data Analysis Grade 6 Grade 7 Grade 8 Grade 12 Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills Building upon knowledge and skills gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the gained in preceding grades, by the end of Grade 6, students will: end of Grade 7, students will: end of Grade 8, students will: end of Grade 12, students will: 1. Collect, generate, organize, 1. Use surveys and sampling and display data. techniques to generate data and draw conclusions about large groups. • Data generated from surveys • Advantages/disadvantages of sample selection methods (e.g., convenience sampling, responses to survey, random sampling) 2. Read, interpret, select, 1. Select and use appropriate 1. Select and use appropriate 2. Evaluate the use of data in construct, analyze, generate representations for sets of representations for sets of real-world contexts. questions about, and draw data, and measures of central data, and measures of central inferences from displays of tendency (mean, median, and tendency (mean, median, and data. mode). mode). • Bar graph, line graph, circle • Type of display most • Type of display most • Accuracy and reasonableness graph, table, histogram appropriate for given data appropriate for given data of conclusions drawn • Range, median, and mean • Box-and-whisker plot, upper • Box-and-whisker plot, upper • Correlation vs. causation quartile, lower quartile quartile, lower quartile • Scatter plot • Scatter plot • Calculators and computers • Calculators and computer • Calculators and computer • Bias in conclusions drawn used to record and process used to record and process used to record and process (e.g., influence of how data information information information is displayed) • Finding the median and • Statistical claims based on mean (weighted average) sampling using frequency data • Effect of additional data on measures of central tendency 3. Estimate lines of best fit and 4. Estimate or determine lines of use them to interpolate within best fit (or curves of best fit if the range of the data. appropriate) with technology, and use them to interpolate within the range of the data. 3. Respond to questions about 2. Make inferences and 2. Make inferences and 5. Analyze data using technology, data, generate their own formulate and evaluate formulate and evaluate and use statistical terminology questions and hypotheses, and arguments based on displays arguments based on displays to describe conclusions. formulate strategies for and analysis of data. and analysis of data sets. • Measures of dispersion: variance, answering their questions and standard deviation, outliers testing their hypotheses. 4. Use surveys and sampling • Correlation coefficient techniques to generate data • Normal distribution (e.g., approx- [Interpreting and using graphic sources of information such as and draw conclusions about imately 95% of the sample lies maps, graphs, timelines, or tables to address research questions large groups. between two standard deviations is included in language arts literacy indicator 3.1.6 H 4] on either side of the mean) 3. Design a statistical experiment, conduct the experiment, and interpret and communicate the outcome. 6. Distinguish between randomized experiments and observational studies. Adopted January 9, 2008 214 Adopted January 9, 2008 Probability. Students need to understand the fundamental concepts of probability so that they can interpret weather forecasts, avoid unfair games of chance, and make informed decisions about medical treatments whose success rate is provided in terms of percentages. They should regularly be engaged in predicting and determining probabilities, often based on experiments (like flipping a coin 100 times), but eventually based on theoretical discussions of probability that make use of systematic counting strategies. High school students should use probability models and solve problems involving compound events and sampling. 4.4.2 B. Probability 4.4.3 B. Probability 4.4.4 B. Probability 4.4.5 B. Probability Grade 2 Grade 3 Grade 4 Grade 5 1. Use chance devices like 1. Use everyday events and 1. Use everyday events and 3. Model situations involving spinners and dice to explore chance devices, such as dice, chance devices, such as dice, probability using simulations concepts of probability. coins, and unevenly divided coins, and unevenly divided (with spinners, dice) and spinners, to explore concepts spinners, to explore concepts theoretical models. of probability. of probability. • Certain, impossible • Likely, unlikely, certain, • Likely, unlikely, certain, impossible impossible, improbable, fair, unfair • More likely, less likely, • More likely, less likely, • More likely, less likely, equally likely equally likely equally likely No Associated Preschool Learning Expectations • Probability of tossing “heads” does not depend on outcomes of previous tosses 2. Provide probability of specific 2. Determine probabilities of 1. Determine probabilities of outcomes. simple events based on events. • Probability of getting equally likely outcomes and • Event, probability of an event specific outcome when coin express them as fractions. • Probability of certain event is is tossed, when die is rolled, 1 and of impossible event is 0 when spinner is spun (e.g., if spinner has five equal sectors, then probability of getting a particular sector is one out of five) • When picking a marble from 2. Predict probabilities in a 3. Predict probabilities in a 2. Determine probability using a bag with three red marbles variety of situations (e.g., variety of situations (e.g., intuitive, experimental, and and four blue marbles, the given the number of items of given the number of items of theoretical methods (e.g., probability of getting a red each color in a bag, what is each color in a bag, what is using model of picking items marble is three out of seven the probability that an item the probability that an item of different colors from a picked will have a particular picked will have a particular bag). color). color). • Given numbers of various • What students think will • What students think will types of items in a bag, what happen (intuitive) happen (intuitive) is the probability that an item • Collect data and use that data • Collect data and use that data of one type will be picked to predict the probability to predict the probability • Given data obtained (experimental) (experimental) experimentally, what is the • Analyze all possible outcomes likely distribution of items in to find the probability the bag (theoretical) Adopted January 9, 2008 215 Adopted January 9, 2008 4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS 4.4.6 B. Probability 4.4.7 B. Probability 4.4.8 B. Probability 4.4.12 B. Probability Grade 6 Grade 7 Grade 8 Grade 12 4. Model situations involving 2. Model situations involving 4. Model situations involving 3. Model situations involving probability using simulations probability with simulations probability with simulations probability with simulations (with spinners, dice) and (using spinners, dice, (using spinners, dice, (using spinners, dice, theoretical models calculators and computers) calculators and computers) calculators and computers) and theoretical models. and theoretical models. and theoretical models, and solve problems using these models. • Frequency, • Frequency, relative frequency relative frequency 1. Determine probabilities of 1. Interpret probabilities as 1. Interpret probabilities as 6. Understand and use the “law events. ratios, percents, and decimals. ratios, percents, and decimals. of large numbers” (that • Event, complementary event, experimental results tend to probability of an event approach theoretical • Multiplication rule for probabilities after a large probabilities number of trials). • Probability of certain event is 1 and of impossible event is 0 • Probabilities of event and complementary event add up to 1 2. Determine probability using 3. Estimate probabilities and 5. Estimate probabilities and 5. Estimate probabilities and intuitive, experimental, and make predictions based on make predictions based on make predictions based on theoretical methods (e.g., experimental and theoretical experimental and theoretical experimental and theoretical using model of picking items probabilities. probabilities. probabilities. of different colors from a bag). • Given numbers of various types of items in a bag, what is the probability that an item of one type will be picked • Given data obtained experimentally, what is the likely distribution of items in the bag 3. Explore compound events. 2. Determine probabilities of compound events. 3. Explore the probabilities of 4. Determine probabilities in conditional events (e.g., if complex situations. there are seven marbles in a • Conditional events bag, three red and four green, • Complementary events what is the probability that • Dependent and independent two marbles picked from the events bag, without replacement, are both red). 5. Recognize and understand the 4. Play and analyze probability- 6. Play and analyze probability- 1. Calculate the expected value connections among the based games, and discuss the based games, and discuss the of a probability-based game, concepts of independent concepts of fairness and concepts of fairness and given the probabilities and outcomes, picking at random, expected value. expected value. payoffs of the various and fairness. outcomes, and determine whether the game is fair. 2. Use concepts and formulas of area to calculate geometric probabilities. Adopted January 9, 2008 216 Adopted January 9, 2008 Discrete Mathematics—Systematic Listing and Counting. Development of strategies for listing and counting can progress through all grade levels, with middle and high school students using the strategies to solve problems in probability. Primary students, for example, might find all outfits that can be worn using two coats and three hats; middle school students might systematically list and count the number of routes from one site on a map to another; and high school students might determine the number of three- person delegations that can be selected from their class to visit the mayor. Preschool Learning 4.4.2C. Discrete Mathematics- 4.4.3C. Discrete Mathematics- 4.4.4C. Discrete Mathematics- 4.4.5C. Discrete Mathematics- Expectations Systematic Listing and Counting Systematic Listing and Counting Systematic Listing and Counting Systematic Listing and Counting Grade 2 Grade 3 Grade 4 Grade 5 3.1 Sorts objects into groups (e.g., separate basket of 1. Sort and classify objects 1. Represent and classify data 1. Represent and classify data [Classifying data cam be related collected items into piles of according to attributes. according to attributes, according to attributes, to classifying organisms, as in pinecones, acorns and twigs). such as shape or color, and such as shape or color, and science indicator 5.5.4 B 1, or [Classifying objects is included relationships. relationships. food groups, as in Preschool in Expectation 3.2 below] Health, Safety and Physical 3.3 Describes an object by charac- Education Expectation 1.1] • Venn diagrams • Venn diagrams • Venn diagrams teristics it does or does not possess (e.g., "This button doesn't have holes."). 3.4 Seriates objects according to • Numerical and • Numerical and various properties including size, alphabetical order alphabetical order number, length, heaviness, texture (rough to smooth) or loudness. [Counting is included in 2. Generate all possibilities 2. Represent all possibilities 2. Represent all possibilities 1. Solve counting problems Preschool Mathematics in simple counting for a simple counting for a simple counting and justify that all Expectations 1.3 through 1.6 situations (e.g., all outfits situation in an organized situation in an organized possibilities have been and 1.8 above] involving two shirts and way and draw conclusions way and draw conclusions enumerated without three pants). from this representation. from this representation. duplication. • Organized lists, charts • Organized lists, charts, • Organized lists, charts, tree diagrams tree diagrams, tables 3.2 Classifies objects by sorting • Dividing into categories them into subgroups by one or (e.g., to find the total more attributes (e.g., sorting number of rectangles in counting bears by color into trays, a grid, find the number separating a mixture of beans by of rectangles of each individual size and shape). size and add the results) 2. Explore the multiplication principle of counting in simple situations by representing all possibilities in an organized way (e.g., you can make 3 x 4 = 12 outfits using 3 shirts and 4 skirts). Adopted January 9, 2008 217 Adopted January 9, 2008 4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS 4.4.6 C. Discrete Mathematics- 4.4.7 C. Discrete Mathematics- 4.4.8 C. Discrete Mathematics- 4.4.12C. Discrete Mathematics- Systematic Listing and Counting Systematic Listing and Counting Systematic Listing and Counting Systematic Listing and Counting Grade 6 Grade 7 Grade 8 Grade 12 [Venn diagrams are included in indicators 4.4.6 C 1, 4.4.7 C 2 , and 4.4.8 C 2 below.] 1. Solve counting problems and 2. Explore counting problems 2. Explore counting problems justify that all possibilities involving Venn diagrams with involving Venn diagrams with have been enumerated three attributes (e.g., there are 15, three attributes (e.g., there are 15, without duplication. 20, and 25 students respectively 20, and 25 students respectively in the chess club, the debating in the chess club, the debating • Organized lists, charts, tree team, and the engineering team, and the engineering diagrams, tables society; how many different society; how many different students belong to the three clubs students belong to the three clubs • Venn diagrams if there are 6 students in chess if there are 6 students in chess and debating, 7 students in chess and debating, 7 students in chess [Venn diargrams are and engineering, 8 students in and engineering, 8 students in introduced in 4.4.2 C 1.] debating and engineering, and 2 debating and engineering, and 2 students in all three?). students in all three?). 2. Apply the multiplication 1. Apply the multiplication 1. Apply the multiplication 2. Apply the multiplication rule principle of counting. principle of counting. principle of counting. of counting in complex • Simple situations (e.g., you • Permutations: ordered • Permutations: ordered situations, recognize the can make 3 x 4 = 12 outfits situations with replacement situations with replacement difference between situations using 3 shirts and 4 skirts). (e.g., number of possible (e.g., number of possible with replacement and without • Number of ways a specified license plates) vs. ordered license plates) vs. ordered replacement, and recognize number of items can be the difference between situations without situations without arranged in order (concept of permutation) replacement (e.g., number replacement (e.g., number ordered and unordered • Number of ways of selecting a of possible slates of 3 class of possible slates of 3 class counting situations. slate of officers from a class officers from a 23 student officers from a 23 student (e.g., if there are 23 students class) class) and 3 officers, the number is • Factorial notation 23 x 22 x 21) 3. List the possible combinations • Concept of combinations 1. Calculate combinations with of two elements chosen from (e.g., number of possible replacement (e.g., the number a given set (e.g., forming a delegations of 3 out of 23 of possible ways of tossing a committee of two from a students) coin 5 times and getting 3 group of 12 students, finding heads) and without how many handshakes there replacement (e.g., number of will be among ten people if possible delegations of 3 out everyone shakes each other of 23 students). person’s hand once). 3. Apply techniques of 3. Apply techniques of 3. Justify solutions to counting systematic listing, counting, systematic listing, counting, problems. and reasoning in a variety of and reasoning in a variety of different contexts. different contexts. [Recognizing, describing, 4. Recognize and explain relationships involving and extending recursive combinations and Pascal’s patterns, including Pascal’s Triangle, and apply those Triangle, is included in methods to situations indicator 4.3.6 A 1.] involving probability. Adopted January 9, 2008 218 Adopted January 9, 2008 Discrete Mathematics—Vertex-Edge Graphs and Algorithms. Vertex-edge graphs, consisting of dots (vertices) and lines joining them (edges), can be used to represent and solve problems based on real-world situations. Students should learn to follow and devise lists of instructions, called “algorithms,” and use algorithmic thinking to find the best solution to problems like those involving vertex-edge graphs, but also to solve other problems. Preschool Learning 4.4.2D. Discrete Mathematics- 4.4.3D. Discrete Mathematics- 4.4.4D. Discrete Mathematics- 4.4.5D. Discrete Mathematics- Expectations Vertex-Edge Graphs and Algorithms Vertex-Edge Graphs and Algorithms Vertex-Edge Graphs and Algorithms Vertex-Edge Graphs and Algorithms Grade 2 Grade 3 Grade 4 Grade 5 4.1 Starts and stops on a signal 1. Follow simple sets of 1. Follow, devise, and 1. Follow, devise, and (e.g., freezing in position directions (e.g., from one describe practical sets of describe practical sets of when the music stops). location to another, or directions (e.g., to add directions (e.g., to add [Following oral directions that involve from a recipe). two 2-digit numbers). two 2-digit numbers). several actions is included in Preschool Language Arts Literacy Expectation 1.1] 3. Play simple two-person 2. Play two-person games 1. Devise strategies for winning games (e.g., tic-tac-toe) and devise strategies for simple games (e.g., start with and informally explore winning the games (e.g., two piles of objects, each of the idea of what the “make 5" where players two players in turn removes outcome should be. alternately add 1 or 2 and any number of objects from a the person who reaches 5, single pile, and the person to [According to N.J.S.A. 18A:35-4.16, “Each board of education or another designated take the last group of objects may offer instruction in chess during the second grade for number, is the winner). wins) and express those pupils in gifted and talented and special education programs.” * strategies as sets of directions. 4. Explore concrete models 2. Explore vertex-edge 3. Explore vertex-edge of vertex-edge graphs (e.g. graphs. graphs and tree diagrams. vertices as “islands” and • Vertex, edge • Vertex, edge, edges as “bridges”). neighboring/adjacent, number of neighbors • Paths from one vertex to • Path • Path, circuit (i.e., path another that ends at its starting point) 2. Color simple maps with a 3. Find the smallest number 4. Find the smallest number small number of colors. of colors needed to color of colors needed to color a map. a map or a graph. * [N.J.S.A. 18A:35-4.15a declares that: ”chess increases strategic thinking skills, stimulates intellectual creativity, and improves problem-solving ability while raising self esteem.”] Adopted January 9, 2008 219 Adopted January 9, 2008 4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS 4.4.6D. Discrete Mathematics- 4.4.7D. Discrete Mathematics- 4.4.8D. Discrete Mathematics- 4.4.12D. Discrete Mathematics- Vertex-Edge Graphs and Algorithms Vertex-Edge Graphs and Algorithms Vertex-Edge Graphs and Algorithms Vertex-Edge Graphs and Algorithms Grade 6 Grade 7 Grade 8 Grade 12 1. Devise strategies for winning simple games (e.g., start with two piles of objects, each of two players in turn removes any number of objects from a single pile, and the person to take the last group of objects wins) and express those strategies as sets of directions. 2. Analyze vertex-edge graphs and tree diagrams. • Can a picture or a vertex-edge graph be drawn with a single line? (degree of vertex) • Can you get from any vertex to any other vertex? (connectedness) 3. Use vertex-edge graphs to find 1. Use vertex-edge graphs to 1. Use vertex-edge graphs and 1. Use vertex-edge graphs and solutions to practical represent and find solutions to algorithmic thinking to algorithmic thinking to problems. practical problems. represent and find solutions to represent and solve practical practical problems. problems. • Delivery route that stops at • Finding the shortest network • Finding the shortest network • Circuits that include every specified sites but involves connecting specified sites connecting specified sites edge in a graph least travel • Finding a minimal route that • Circuits that include every includes every street (e.g., for trash pick-up) vertex in a graph • Finding the shortest route on a • Scheduling problems (e.g., • Shortest route from one site • Finding the shortest route on a map from one site to another when project meetings on a map to another map from one site to another • Finding the shortest circuit on should be scheduled to • Finding the shortest circuit on a map that makes a tour of avoid conflicts) using graph a map that makes a tour of specified sites coloring specified sites • Limitations of computers • Applications to science (e.g., the number of routes for a (e.g., who-eats-whom delivery truck visiting n sites is graphs, genetic trees, n!, so finding the shortest circuit by examining all circuits would molecular structures) overwhelm the capacity of any computer, now or in the future, even if n is less than 100) 2. Explore strategies for making fair decisions. • Combining individual preferences into a group decision (e.g., determining winner of an election or selection process) • Determining how many Student Council representatives each class (9th, 10th, 11th, and 12th grade) gets when the classes have unequal sizes (apportionment) These topics provide students with insight into how mathematics is used by decision-makers in our society, and with important tools for modeling a variety of real-world situations. Students will better understand and interpret the vast amounts of quantitative data that they are exposed to daily, and they will be able to judge the validity of data-supported arguments. Adopted January 9, 2008 220 Adopted January 9, 2008 Math Websites Let’s Get Close - Students practice improper fractions and mixed numbers by rolling dice http://www.math.montana.edu/mathed/distance/capstone/kershner/activities/get_cl ose.html Multiplication Machine - Students can practice multiplying from easy to megahard http://www.amblesideprimary.com/ambleweb/mentalmaths/testtest.html Sum Sense Division - Students have to complete division sentences is an allotted amount of time http://www.oswego.org/ocsd-web/games/SumSense/sumdiv.html Sum sense Multiplication - Students have to complete division sentences is an allotted amount of time http://www.oswego.org/ocsd-web/games/SumSense/summulti.html A great website with lots of Number Sense Interactive Games http://edweb.tusd.k12.az.us/ekowalcz/math/elementary_web_sites.htm 221

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